1 00:00:00,990 --> 00:00:02,475 [SQUEAKING] 2 00:00:02,475 --> 00:00:03,960 [RUSTLING] 3 00:00:03,960 --> 00:00:10,500 [CLICKING] 4 00:00:10,500 --> 00:00:12,000 NANCY KANWISHER: All right, so we're 5 00:00:12,000 --> 00:00:13,587 going to talk about number. 6 00:00:13,587 --> 00:00:16,170 I got a little carried away with the behavioral work on number 7 00:00:16,170 --> 00:00:18,623 because I just think it is so awesome. 8 00:00:18,623 --> 00:00:20,790 And I think it's, frankly, a little more interesting 9 00:00:20,790 --> 00:00:22,040 than a lot of the neural work. 10 00:00:22,040 --> 00:00:25,740 So this is going to be sort of a behaviorally heavy lecture. 11 00:00:25,740 --> 00:00:30,390 But let's start by thinking about why we have number 12 00:00:30,390 --> 00:00:32,009 and what we use it for. 13 00:00:32,009 --> 00:00:33,900 And the first thing to realize is 14 00:00:33,900 --> 00:00:38,700 that we use concepts of number and quantity like all the time. 15 00:00:38,700 --> 00:00:41,182 Most obviously, if you're, say, getting change at a store. 16 00:00:41,182 --> 00:00:43,390 I guess that doesn't really happen very much anymore. 17 00:00:43,390 --> 00:00:45,510 People are going to forget how to subtract because they just 18 00:00:45,510 --> 00:00:47,370 put their credit card or bump their phone 19 00:00:47,370 --> 00:00:48,250 or whatever they do. 20 00:00:48,250 --> 00:00:50,610 But anyway, it used to be that you handed 21 00:00:50,610 --> 00:00:53,190 over this stuff called money and that coins came back 22 00:00:53,190 --> 00:00:56,280 and that was the subtraction involved. 23 00:00:56,280 --> 00:00:59,970 We use it to tell time or to fail to tell time, 24 00:00:59,970 --> 00:01:02,310 as in my case this morning. 25 00:01:02,310 --> 00:01:04,410 To choose the larger of two objects, 26 00:01:04,410 --> 00:01:06,250 that's a continuous idea of quantity, 27 00:01:06,250 --> 00:01:09,030 not a discrete idea of number. 28 00:01:09,030 --> 00:01:11,880 To choose the shortest line at a grocery store, right, 29 00:01:11,880 --> 00:01:16,110 and all of those kinds of things are comparing quantities. 30 00:01:16,110 --> 00:01:20,130 And we also take these basic ideas of number and quantity 31 00:01:20,130 --> 00:01:22,140 and we build on them in modern societies 32 00:01:22,140 --> 00:01:26,940 to do all kinds of amazing things like engineering. 33 00:01:26,940 --> 00:01:29,760 Like all of modern science is highly quantitative, 34 00:01:29,760 --> 00:01:32,440 like all of computer science. 35 00:01:32,440 --> 00:01:35,670 And so these are really fundamental ideas. 36 00:01:35,670 --> 00:01:40,770 And animals, it turns out, are capable of mastering 37 00:01:40,770 --> 00:01:45,030 very simple but sophisticated understandings of number 38 00:01:45,030 --> 00:01:49,650 and even arithmetic computations. 39 00:01:49,650 --> 00:01:54,540 They can learn about order and number and quantity. 40 00:01:54,540 --> 00:01:59,410 OK, and they need to for lots of reasons. 41 00:01:59,410 --> 00:02:03,210 So here, just a brief overview of some of the situations 42 00:02:03,210 --> 00:02:05,970 where animals need concepts of number and quantity 43 00:02:05,970 --> 00:02:07,770 in the wild. 44 00:02:07,770 --> 00:02:10,080 Foraging, right, so animals spend a lot of time 45 00:02:10,080 --> 00:02:11,922 rooting around for food over here, 46 00:02:11,922 --> 00:02:13,380 rooting around for food over there, 47 00:02:13,380 --> 00:02:16,650 deciding when to keep rooting around here despite diminishing 48 00:02:16,650 --> 00:02:20,910 returns and go somewhere else where there's unknown pay off, 49 00:02:20,910 --> 00:02:21,960 unknown amounts of food. 50 00:02:21,960 --> 00:02:25,460 So that's a whole math of foraging behavior. 51 00:02:25,460 --> 00:02:28,400 OK, so that deals with the rate of return 52 00:02:28,400 --> 00:02:31,490 of the food at each location and the amount and the quality. 53 00:02:31,490 --> 00:02:36,180 And you can imagine a whole math to optimize the amount of food. 54 00:02:36,180 --> 00:02:38,960 They also need to know about number and quantity 55 00:02:38,960 --> 00:02:42,620 when they form teams, which many animals across taxa 56 00:02:42,620 --> 00:02:44,550 do in different ways. 57 00:02:44,550 --> 00:02:48,050 So schooling fish can quickly pick out 58 00:02:48,050 --> 00:02:52,160 the more numerous school of fish to join. 59 00:02:52,160 --> 00:02:54,987 And that's what they want to do because your statistics are 60 00:02:54,987 --> 00:02:56,570 better if they're a predator if you're 61 00:02:56,570 --> 00:02:59,600 in the larger school than the smaller school, right? 62 00:03:02,170 --> 00:03:05,200 So your chance of getting eaten is reduced just 63 00:03:05,200 --> 00:03:07,910 dividing by the number of options. 64 00:03:07,910 --> 00:03:09,670 And then there's all kinds of animals 65 00:03:09,670 --> 00:03:13,630 that take into account the size of groups of their own species 66 00:03:13,630 --> 00:03:19,870 or other species when making decisions about how far to run 67 00:03:19,870 --> 00:03:23,380 or who to chase or who to predate on 68 00:03:23,380 --> 00:03:27,280 or who's at risk of predating upon you. 69 00:03:27,280 --> 00:03:30,160 So lions hunt in teams. 70 00:03:30,160 --> 00:03:31,460 And they have to work together. 71 00:03:31,460 --> 00:03:34,060 They have actually whole strategic situation 72 00:03:34,060 --> 00:03:36,700 where different lions play different parts like a football 73 00:03:36,700 --> 00:03:38,260 game. 74 00:03:38,260 --> 00:03:41,500 And they have to decide which groups of predators 75 00:03:41,500 --> 00:03:45,520 to take on based on numerical advantage. 76 00:03:45,520 --> 00:03:49,390 And my favorite is the n plus one frog, the Tungara 77 00:03:49,390 --> 00:03:54,890 frog that lives in the rainforest in Puerto Rico. 78 00:03:54,890 --> 00:03:58,420 And it literally one ups other frogs, the males 79 00:03:58,420 --> 00:04:01,210 do in trying to impress the females. 80 00:04:01,210 --> 00:04:05,320 And so what happens is that one frog will start calling out. 81 00:04:05,320 --> 00:04:07,570 One male frog will start calling out trying 82 00:04:07,570 --> 00:04:10,090 to sound all hot to the gals. 83 00:04:10,090 --> 00:04:13,690 And then another frog will one up him by doing that call 84 00:04:13,690 --> 00:04:17,050 but elaborating on it by adding an extra call 85 00:04:17,050 --> 00:04:18,970 or an extra component. 86 00:04:18,970 --> 00:04:21,955 So for example-- 87 00:04:21,955 --> 00:04:27,180 [FROG CALL] OK, so that's one dude calling out. 88 00:04:27,180 --> 00:04:32,766 And not to be outdone, the next guy calls back. 89 00:04:32,766 --> 00:04:37,780 [FROG CALL] And apparently, if you follow these guys, they 90 00:04:37,780 --> 00:04:42,760 pretty systematically add one to the previous frog's call, 91 00:04:42,760 --> 00:04:45,220 right, up to a point. 92 00:04:45,220 --> 00:04:47,800 The point being approximately four. 93 00:04:47,800 --> 00:04:51,010 So it's not like 100 and 101. 94 00:04:51,010 --> 00:04:53,950 But it happens. 95 00:04:53,950 --> 00:04:56,680 OK, so that's just a broad overview of some of the cases 96 00:04:56,680 --> 00:05:00,700 that understandings of number and quantity 97 00:05:00,700 --> 00:05:04,370 arise in natural environments without training. 98 00:05:04,370 --> 00:05:07,270 So we want to know how is all this computed in the mind 99 00:05:07,270 --> 00:05:09,020 and brain. 100 00:05:09,020 --> 00:05:12,430 And so what are the foremost thinkers on this topic 101 00:05:12,430 --> 00:05:15,260 is Stan Dehaene, shown here. 102 00:05:15,260 --> 00:05:20,620 And he wrote in a very widely cited book, actually 103 00:05:20,620 --> 00:05:24,400 review article and book quite a while ago, 20 years ago, 104 00:05:24,400 --> 00:05:28,210 he said, animals, young infants, and adult humans 105 00:05:28,210 --> 00:05:31,450 possess a biologically determined, 106 00:05:31,450 --> 00:05:35,240 domain-specific representation of number. 107 00:05:35,240 --> 00:05:38,488 So this is a very kind of extreme, hardcore claim. 108 00:05:38,488 --> 00:05:40,030 We will see at the end of the lecture 109 00:05:40,030 --> 00:05:42,700 that he has backed off that claim. 110 00:05:42,700 --> 00:05:46,090 OK, so a couple of things, biologically determined, 111 00:05:46,090 --> 00:05:49,720 he's kind of implying innate, right? 112 00:05:49,720 --> 00:05:52,690 Domain-specific, I've avoided this phrase, 113 00:05:52,690 --> 00:05:54,880 for the most part, because it's kind of like jargon 114 00:05:54,880 --> 00:05:55,600 gobbledygook. 115 00:05:55,600 --> 00:05:58,282 But it's actually so entrenched in our field 116 00:05:58,282 --> 00:05:59,740 that it's worth knowing what it is. 117 00:05:59,740 --> 00:06:02,860 Domain-specific is just this idea of functional specificity 118 00:06:02,860 --> 00:06:04,120 that I've been talking about. 119 00:06:04,120 --> 00:06:06,940 But you can apply it to not just a piece of brain like, 120 00:06:06,940 --> 00:06:09,550 does this patch of brain process only faces? 121 00:06:09,550 --> 00:06:12,160 You can also apply it to a mental process 122 00:06:12,160 --> 00:06:14,500 even if you don't know what its actual brain basis is. 123 00:06:14,500 --> 00:06:17,350 So do we have special mental machinery 124 00:06:17,350 --> 00:06:18,820 for thinking about numbers that's 125 00:06:18,820 --> 00:06:22,420 distinct from our machinery for face recognition or navigation 126 00:06:22,420 --> 00:06:24,100 or language or whatever else? 127 00:06:24,100 --> 00:06:26,410 OK, so that's what domain-specific means. 128 00:06:26,410 --> 00:06:27,970 And it's worth knowing because you'll 129 00:06:27,970 --> 00:06:30,430 encounter it in other contexts. 130 00:06:30,430 --> 00:06:35,410 OK, so in more detail, Stan says, 131 00:06:35,410 --> 00:06:38,050 a specific neural substrate, located 132 00:06:38,050 --> 00:06:40,900 in the left intraparietal area, is 133 00:06:40,900 --> 00:06:43,810 associated with knowledge of numbers and their relations, 134 00:06:43,810 --> 00:06:46,090 which he defines as number sense. 135 00:06:46,090 --> 00:06:47,950 The number domain is a prime example 136 00:06:47,950 --> 00:06:51,190 where strong evidence points to an evolutionary endowment 137 00:06:51,190 --> 00:06:54,250 of abstract, domain-specific knowledge in the brain 138 00:06:54,250 --> 00:06:56,620 because there are parallels between number processing 139 00:06:56,620 --> 00:06:58,720 in animals and humans. 140 00:06:58,720 --> 00:07:00,790 Again, kind of hardcore claims. 141 00:07:00,790 --> 00:07:04,120 Not just is there this so he doesn't quite say innate, 142 00:07:04,120 --> 00:07:05,770 but he's strongly implying innate. 143 00:07:05,770 --> 00:07:08,380 I mean, that's evolutionary endowment, that 144 00:07:08,380 --> 00:07:10,180 basically means innate, right? 145 00:07:10,180 --> 00:07:13,240 It's an evolved ability that lives in a particular part 146 00:07:13,240 --> 00:07:14,200 of the brain. 147 00:07:14,200 --> 00:07:14,980 OK? 148 00:07:14,980 --> 00:07:16,660 So who would a thunk, right? 149 00:07:16,660 --> 00:07:17,243 Number, right? 150 00:07:17,243 --> 00:07:19,660 You think of number as something you get taught in school. 151 00:07:19,660 --> 00:07:21,220 But no, he's saying it's really part 152 00:07:21,220 --> 00:07:23,000 of your biological endowment. 153 00:07:23,000 --> 00:07:25,480 It has a particular brain region. 154 00:07:25,480 --> 00:07:28,990 And all of that may be if not completely independent, 155 00:07:28,990 --> 00:07:32,370 it may exist without explicit training. 156 00:07:32,370 --> 00:07:32,870 OK? 157 00:07:32,870 --> 00:07:34,670 So that's quite a claim. 158 00:07:34,670 --> 00:07:38,200 So what does number sense mean exactly? 159 00:07:38,200 --> 00:07:41,910 Well, what Stan and others in the field mean by number sense, 160 00:07:41,910 --> 00:07:43,180 it's a bunch of things. 161 00:07:43,180 --> 00:07:46,085 First of all, the idea that for human adults 162 00:07:46,085 --> 00:07:48,210 to have number sense, that means they can represent 163 00:07:48,210 --> 00:07:52,350 large numerical magnitudes without verbal counting, right? 164 00:07:52,350 --> 00:07:53,850 So counting is an interesting thing. 165 00:07:53,850 --> 00:07:55,892 But we're going to leave it aside for the moment. 166 00:07:55,892 --> 00:07:58,200 Number sense is a more general idea 167 00:07:58,200 --> 00:08:01,200 that's going to apply to animals and infants 168 00:08:01,200 --> 00:08:02,815 without explicit counting. 169 00:08:02,815 --> 00:08:05,190 OK, so you can have some way of representing that there's 170 00:08:05,190 --> 00:08:06,065 a lot of things here. 171 00:08:06,065 --> 00:08:08,970 And there's fewer things there. 172 00:08:08,970 --> 00:08:10,470 Second of all, these representations 173 00:08:10,470 --> 00:08:12,870 are approximate. 174 00:08:12,870 --> 00:08:15,510 And the ability to discriminate two of them 175 00:08:15,510 --> 00:08:17,910 depends on the ratio of those two, 176 00:08:17,910 --> 00:08:20,160 not the absolute difference. 177 00:08:20,160 --> 00:08:22,620 OK, and I'll show you in more detail what I mean by that. 178 00:08:22,620 --> 00:08:24,450 It's a deep fact about number sense 179 00:08:24,450 --> 00:08:28,200 and actually all of perception, pretty much. 180 00:08:28,200 --> 00:08:31,800 Further, the idea is that these representations are abstract. 181 00:08:31,800 --> 00:08:35,100 They're not just, say, a particular visual form. 182 00:08:35,100 --> 00:08:38,169 Like approximately 13 looks like this. 183 00:08:38,169 --> 00:08:38,669 No. 184 00:08:38,669 --> 00:08:41,640 They're going to generalize across modality, OK, 185 00:08:41,640 --> 00:08:44,620 and space and time. 186 00:08:44,620 --> 00:08:50,100 Next, these mental representations of number 187 00:08:50,100 --> 00:08:51,870 can be used in operations. 188 00:08:51,870 --> 00:08:55,260 Even without counting and being explicitly informed, 189 00:08:55,260 --> 00:08:57,150 you can add approximate numbers. 190 00:08:57,150 --> 00:08:59,400 You may be thinking, what the hell am I talking about? 191 00:08:59,400 --> 00:09:02,050 But I'll show you in a second. 192 00:09:02,050 --> 00:09:07,650 So for example, I'm going to show you two sets of dots next. 193 00:09:07,650 --> 00:09:09,660 And you're just going to shout out first 194 00:09:09,660 --> 00:09:12,360 if the first set of dots had more, 195 00:09:12,360 --> 00:09:14,310 if there were more dots in the first array 196 00:09:14,310 --> 00:09:16,930 and second if the second array had more dots. 197 00:09:16,930 --> 00:09:17,430 OK, ready? 198 00:09:17,430 --> 00:09:18,750 Here we go. 199 00:09:18,750 --> 00:09:19,880 Boom. 200 00:09:19,880 --> 00:09:22,800 Boom. 201 00:09:22,800 --> 00:09:23,400 Second. 202 00:09:23,400 --> 00:09:23,910 Duh. 203 00:09:23,910 --> 00:09:24,993 OK, let's try another one. 204 00:09:28,980 --> 00:09:29,693 Duh. 205 00:09:29,693 --> 00:09:30,360 And another one. 206 00:09:33,162 --> 00:09:33,990 Uh huh. 207 00:09:33,990 --> 00:09:34,590 Another one. 208 00:09:40,930 --> 00:09:43,270 I noticed the volume decreasing. 209 00:09:43,270 --> 00:09:44,920 And I noticed some hesitancy. 210 00:09:48,950 --> 00:09:51,840 Actually, I'm not sure about that one. 211 00:09:51,840 --> 00:09:56,190 OK, so how did you do that? 212 00:09:56,190 --> 00:09:57,410 What did you do? 213 00:09:57,410 --> 00:10:00,620 Did you go 1, 2, 3, 4, 5? 214 00:10:00,620 --> 00:10:01,120 No. 215 00:10:01,120 --> 00:10:02,860 I tried to do it, so there wasn't time to do that. 216 00:10:02,860 --> 00:10:03,597 How'd you do it? 217 00:10:03,597 --> 00:10:05,680 AUDIENCE: I kind of tried to see like the density, 218 00:10:05,680 --> 00:10:07,110 like how close all the dots were. 219 00:10:07,110 --> 00:10:08,110 NANCY KANWISHER: Mm-hmm. 220 00:10:08,110 --> 00:10:08,590 Mm-hmm. 221 00:10:08,590 --> 00:10:09,673 And did that work for you? 222 00:10:09,673 --> 00:10:11,610 Did that work OK? 223 00:10:11,610 --> 00:10:13,015 AUDIENCE: It seems to be OK. 224 00:10:13,015 --> 00:10:14,890 NANCY KANWISHER: OK, what Jack is pointing to 225 00:10:14,890 --> 00:10:16,515 is a really important thing in thinking 226 00:10:16,515 --> 00:10:18,730 about number, which is that number 227 00:10:18,730 --> 00:10:21,040 is confounded with area extent. 228 00:10:21,040 --> 00:10:23,410 How much total yellow stuff is on the screen? 229 00:10:23,410 --> 00:10:25,460 And it's confounded with density. 230 00:10:25,460 --> 00:10:28,090 And this is a big problem in people who 231 00:10:28,090 --> 00:10:29,500 want to do research on number. 232 00:10:29,500 --> 00:10:31,500 And so what they usually do is you can't totally 233 00:10:31,500 --> 00:10:32,560 unconfound those things. 234 00:10:32,560 --> 00:10:34,850 But you can unconfound them one at a time. 235 00:10:34,850 --> 00:10:37,090 So you can vary the size of the objects. 236 00:10:37,090 --> 00:10:39,230 And you can vary the density across trials. 237 00:10:39,230 --> 00:10:42,343 So no one of those cues will enable you to do it fully. 238 00:10:42,343 --> 00:10:44,260 This example isn't great that way because they 239 00:10:44,260 --> 00:10:47,750 were all the same size, right? 240 00:10:47,750 --> 00:10:50,990 OK, but so the point is, without explicitly counting, 241 00:10:50,990 --> 00:10:52,950 and God knows what you do it, how you do it, 242 00:10:52,950 --> 00:10:55,400 you just feel like you have a sense of roughly how many. 243 00:10:55,400 --> 00:10:56,960 Everybody got that sense? 244 00:10:56,960 --> 00:10:59,150 OK, so that's what we mean by number sense 245 00:10:59,150 --> 00:11:01,310 is that sense that you can just look at something 246 00:11:01,310 --> 00:11:03,290 and have a sense of roughly how many. 247 00:11:03,290 --> 00:11:05,420 Like you don't know if it's 19 or 18, 248 00:11:05,420 --> 00:11:08,720 but you know it's not 13, right? 249 00:11:08,720 --> 00:11:12,080 OK, right. 250 00:11:12,080 --> 00:11:16,400 Oh, and you guys all got quieter when the numbers 251 00:11:16,400 --> 00:11:17,580 got closer together. 252 00:11:17,580 --> 00:11:18,080 OK? 253 00:11:18,080 --> 00:11:20,930 It gets harder when the numbers are closer together. 254 00:11:20,930 --> 00:11:24,657 OK, so in experiments that have quantified this, lots of people 255 00:11:24,657 --> 00:11:25,490 have looked at this. 256 00:11:25,490 --> 00:11:27,450 Here's one that I was involved in way back. 257 00:11:27,450 --> 00:11:30,590 Just like you did, this is the task here 258 00:11:30,590 --> 00:11:32,558 that you guys just did. 259 00:11:32,558 --> 00:11:34,100 And here are some of the data we got. 260 00:11:34,100 --> 00:11:35,433 So let me walk you through this. 261 00:11:35,433 --> 00:11:38,450 This is accuracy on a bunch of different comparisons. 262 00:11:38,450 --> 00:11:43,220 16 dots versus 32 dots, people are pretty much 100% correct. 263 00:11:43,220 --> 00:11:44,060 OK? 264 00:11:44,060 --> 00:11:46,970 This is just normal human adults. 265 00:11:46,970 --> 00:11:48,980 16 versus 24, great. 266 00:11:48,980 --> 00:11:51,020 16 versus 20, pretty good. 267 00:11:51,020 --> 00:11:54,110 16 versus 18, now we're really dropping. 268 00:11:54,110 --> 00:11:56,030 16 versus 17, forget it. 269 00:11:56,030 --> 00:11:57,000 Can't do it. 270 00:11:57,000 --> 00:11:57,500 OK? 271 00:11:57,500 --> 00:12:01,790 So performance falls off as the numbers get closer together. 272 00:12:01,790 --> 00:12:02,300 OK? 273 00:12:02,300 --> 00:12:04,310 So that's sort of intuitive. 274 00:12:04,310 --> 00:12:11,540 But now let's consider these are all comparing to 16. 275 00:12:11,540 --> 00:12:13,000 Here, we compare to eight. 276 00:12:13,000 --> 00:12:14,630 Eight versus 16. 277 00:12:14,630 --> 00:12:15,830 Eight versus 12. 278 00:12:15,830 --> 00:12:16,580 Eight versus 10. 279 00:12:16,580 --> 00:12:17,330 Eight versus nine. 280 00:12:17,330 --> 00:12:20,900 You see the same fall off as the numbers get closer together. 281 00:12:20,900 --> 00:12:21,920 OK? 282 00:12:21,920 --> 00:12:23,490 So far, so good. 283 00:12:23,490 --> 00:12:26,690 But now we can ask, what determines that fall off? 284 00:12:26,690 --> 00:12:30,090 Is it the absolute difference or the ratio? 285 00:12:30,090 --> 00:12:36,110 And the way we tell is we plot the ratio of those two curves, 286 00:12:36,110 --> 00:12:37,640 and we look at performance. 287 00:12:37,640 --> 00:12:41,090 And we see they are spot on top of each other. 288 00:12:41,090 --> 00:12:43,670 That tells us that it is not the absolute difference that 289 00:12:43,670 --> 00:12:45,440 determines your ability to do this 290 00:12:45,440 --> 00:12:48,030 but the ratio of the numbers of dots. 291 00:12:48,030 --> 00:12:48,530 OK? 292 00:12:48,530 --> 00:12:50,690 It's sort of intuitive, right? 293 00:12:50,690 --> 00:12:53,510 But it's amazing how clear the result is. 294 00:12:53,510 --> 00:12:55,040 Everybody get that? 295 00:12:55,040 --> 00:12:58,670 OK, so this is a really deep fundamental fact 296 00:12:58,670 --> 00:13:02,480 about perceiving approximate number. 297 00:13:02,480 --> 00:13:04,130 And it's actually, more generally, 298 00:13:04,130 --> 00:13:05,690 a fact about perception. 299 00:13:05,690 --> 00:13:07,130 It's called Weber's law. 300 00:13:07,130 --> 00:13:09,980 And it just means that the discriminability 301 00:13:09,980 --> 00:13:12,470 of, in this case, two numbers, two numerosities, 302 00:13:12,470 --> 00:13:15,530 depends on their ratio, not their absolute difference. 303 00:13:15,530 --> 00:13:20,240 The exact same thing holds for evaluating which of two stimuli 304 00:13:20,240 --> 00:13:23,280 is brighter, which of two objects is heavier, 305 00:13:23,280 --> 00:13:24,980 which of two sounds is louder. 306 00:13:24,980 --> 00:13:26,810 They all follow. 307 00:13:26,810 --> 00:13:28,430 The ability to do that is a function 308 00:13:28,430 --> 00:13:31,490 of the ratio of the, two not the absolute difference. 309 00:13:31,490 --> 00:13:32,550 Yeah? 310 00:13:32,550 --> 00:13:38,385 AUDIENCE: [INAUDIBLE] with the size of the dots? 311 00:13:38,385 --> 00:13:40,010 NANCY KANWISHER: So in this experiment, 312 00:13:40,010 --> 00:13:44,510 we varied the sizes every which way and the density. 313 00:13:44,510 --> 00:13:47,300 As I mentioned before, you can't completely 314 00:13:47,300 --> 00:13:51,710 unconfound both size and density within each trial. 315 00:13:51,710 --> 00:13:53,620 But across trials, you can muck them up. 316 00:13:53,620 --> 00:13:55,370 So you can ask whether people are doing it 317 00:13:55,370 --> 00:13:57,470 by size or by density. 318 00:13:57,470 --> 00:13:58,340 OK? 319 00:13:58,340 --> 00:14:00,690 And we did all that here. 320 00:14:00,690 --> 00:14:02,880 OK, so this is not shocking yet. 321 00:14:02,880 --> 00:14:06,600 It's just kind of a basic, deep, clear fact about whatever 322 00:14:06,600 --> 00:14:08,160 our mental representation of number 323 00:14:08,160 --> 00:14:10,650 is, that it's this approximate thing. 324 00:14:10,650 --> 00:14:11,730 It's pretty good. 325 00:14:11,730 --> 00:14:16,950 And its precision scales with the magnitude. 326 00:14:16,950 --> 00:14:22,560 OK, all right, so this has been quantified in lots and lots 327 00:14:22,560 --> 00:14:23,820 of experiments. 328 00:14:23,820 --> 00:14:25,560 And this is called the Approximate Number 329 00:14:25,560 --> 00:14:28,080 System, or ANS. 330 00:14:28,080 --> 00:14:30,570 And the standard test that's been used in lots of studies 331 00:14:30,570 --> 00:14:34,470 to measure people's kind of number acuity 332 00:14:34,470 --> 00:14:36,603 is a lot like what I just showed you. 333 00:14:36,603 --> 00:14:37,770 You show an array like this. 334 00:14:37,770 --> 00:14:41,340 And you say, are there more yellow dots or blue dots? 335 00:14:41,340 --> 00:14:43,710 And people very quickly say yellow, in this case. 336 00:14:43,710 --> 00:14:45,720 And then you ask for a case like this. 337 00:14:45,720 --> 00:14:47,940 And they're a little slower, right? 338 00:14:47,940 --> 00:14:50,550 And here, you can see that the sizes have changed 339 00:14:50,550 --> 00:14:53,160 and have been orthogonalized. 340 00:14:53,160 --> 00:14:57,510 OK, so the ratio of yellow to blue dots 341 00:14:57,510 --> 00:14:59,610 is called the Weber fraction, right? 342 00:14:59,610 --> 00:15:02,970 This is this idea of Weber's law that determines your accuracy 343 00:15:02,970 --> 00:15:05,700 from just that ratio. 344 00:15:05,700 --> 00:15:08,810 And so you can measure people's Weber fraction, their ability 345 00:15:08,810 --> 00:15:11,582 to do this task, their kind of number precision. 346 00:15:11,582 --> 00:15:13,040 And what you find is, first of all, 347 00:15:13,040 --> 00:15:15,800 that there's very big individual differences. 348 00:15:15,800 --> 00:15:16,790 OK? 349 00:15:16,790 --> 00:15:17,870 Now, this is interesting. 350 00:15:17,870 --> 00:15:20,180 It's like things that we've seen in other domains. 351 00:15:20,180 --> 00:15:22,160 There are very big individual differences 352 00:15:22,160 --> 00:15:24,040 in navigational ability. 353 00:15:24,040 --> 00:15:25,910 There are very big individual differences 354 00:15:25,910 --> 00:15:28,610 in face recognition ability. 355 00:15:28,610 --> 00:15:32,790 And in both of those cases as well, 356 00:15:32,790 --> 00:15:37,100 there are people who are just so bad at it, from an early age, 357 00:15:37,100 --> 00:15:38,390 that it's like a syndrome. 358 00:15:38,390 --> 00:15:42,170 In this case, it's called developmental dyscalculia. 359 00:15:42,170 --> 00:15:44,420 I think I didn't fit it into the navigation lectures. 360 00:15:44,420 --> 00:15:48,200 But there's a whole kind of developmental disability 361 00:15:48,200 --> 00:15:51,440 in navigation that's called developmental topographic 362 00:15:51,440 --> 00:15:52,040 agnosia. 363 00:15:52,040 --> 00:15:55,550 People were just always really awful at knowing 364 00:15:55,550 --> 00:15:57,320 where they are, right? 365 00:15:57,320 --> 00:16:00,560 And I did mention developmental prosopagnosia. 366 00:16:00,560 --> 00:16:03,380 People were just always awful at face recognition. 367 00:16:03,380 --> 00:16:07,040 In each of these cases, in the apparent lack 368 00:16:07,040 --> 00:16:11,900 of any evidence of brain damage and in the absence 369 00:16:11,900 --> 00:16:14,780 of differences in IQ or other abilities. 370 00:16:14,780 --> 00:16:16,910 So it seems like each of those abilities 371 00:16:16,910 --> 00:16:18,500 has a very broad range. 372 00:16:18,500 --> 00:16:20,030 At the bottom end of the range, it's 373 00:16:20,030 --> 00:16:23,390 really kind of affects your life you're so bad. 374 00:16:23,390 --> 00:16:25,028 And it's unrelated to other abilities. 375 00:16:25,028 --> 00:16:26,570 And I think that's pretty interesting 376 00:16:26,570 --> 00:16:28,340 because it goes along with the idea 377 00:16:28,340 --> 00:16:32,090 that those mental abilities are really distinct parts of mind 378 00:16:32,090 --> 00:16:32,930 and brain. 379 00:16:32,930 --> 00:16:34,820 You can have a crappy number sense, 380 00:16:34,820 --> 00:16:38,090 and it doesn't mean that you're bad at other things. 381 00:16:38,090 --> 00:16:39,770 You just have a crappy number sense. 382 00:16:39,770 --> 00:16:42,120 It's a separate system, right? 383 00:16:42,120 --> 00:16:42,620 OK. 384 00:16:45,800 --> 00:16:48,470 Approximate number sense develops slowly. 385 00:16:48,470 --> 00:16:50,210 It's best at age 30. 386 00:16:50,210 --> 00:16:52,010 You guys are still on the upswing. 387 00:16:52,010 --> 00:16:54,860 We won't talk about me. 388 00:16:54,860 --> 00:16:57,560 This is what do we have here? 389 00:16:57,560 --> 00:16:58,950 This is Weber fraction. 390 00:16:58,950 --> 00:17:01,910 So the Weber fraction is what that ratio 391 00:17:01,910 --> 00:17:05,640 needs to be for you to be fairly accurate on whatever criteria 392 00:17:05,640 --> 00:17:06,140 they chose. 393 00:17:06,140 --> 00:17:08,810 And so a small fraction means you're better. 394 00:17:08,810 --> 00:17:10,380 And so it goes down. 395 00:17:10,380 --> 00:17:13,520 And this is age here, best at 30. 396 00:17:13,520 --> 00:17:17,599 And this is reaction time, which goes up for everything. 397 00:17:17,599 --> 00:17:19,420 What a bummer. 398 00:17:19,420 --> 00:17:20,300 Anyway. 399 00:17:20,300 --> 00:17:24,290 Interestingly, early ability with approximate number 400 00:17:24,290 --> 00:17:27,710 on this kind of a test predicts later math ability 401 00:17:27,710 --> 00:17:32,150 with very different kinds of organized math 402 00:17:32,150 --> 00:17:34,560 that you learn in school. 403 00:17:34,560 --> 00:17:36,960 So here's a study that looked at that. 404 00:17:36,960 --> 00:17:39,690 They asked whether this early approximate number 405 00:17:39,690 --> 00:17:42,960 sense is predictive of later arithmetic ability. 406 00:17:42,960 --> 00:17:46,107 And so in this case, they did a task like this. 407 00:17:46,107 --> 00:17:48,690 And their measure, they didn't use the task I just showed you. 408 00:17:48,690 --> 00:17:50,773 This is another thing you can do with little kids. 409 00:17:50,773 --> 00:17:51,990 You just flash this up. 410 00:17:51,990 --> 00:17:54,120 And you just ask them, how many dots are there? 411 00:17:54,120 --> 00:17:55,800 And they have to say four, right? 412 00:17:55,800 --> 00:17:57,420 And you just measure reaction time. 413 00:17:57,420 --> 00:17:58,710 It's pretty basic. 414 00:17:58,710 --> 00:18:00,540 OK? 415 00:18:00,540 --> 00:18:04,080 And so then what you do is you run this on kindergarteners. 416 00:18:04,080 --> 00:18:06,810 And you define groups that are slow, medium, 417 00:18:06,810 --> 00:18:08,820 or fast at this task. 418 00:18:08,820 --> 00:18:10,120 OK? 419 00:18:10,120 --> 00:18:12,360 So then you follow them. 420 00:18:12,360 --> 00:18:14,430 And you look at them later, in this case, 421 00:18:14,430 --> 00:18:18,880 at age nine and six years. 422 00:18:18,880 --> 00:18:23,310 And what you see is, even these older kids, who 423 00:18:23,310 --> 00:18:28,330 are defined by the slow, medium, or fast group in kindergarten, 424 00:18:28,330 --> 00:18:31,470 this is now their accuracy at arithmetic tasks 425 00:18:31,470 --> 00:18:33,740 four years later. 426 00:18:33,740 --> 00:18:35,180 Yeah? 427 00:18:35,180 --> 00:18:38,570 So it's not just some weird little task 428 00:18:38,570 --> 00:18:42,140 that psychophysicists made up to measure God knows what. 429 00:18:42,140 --> 00:18:46,200 It's predictive of your later arithmetic ability. 430 00:18:46,200 --> 00:18:46,920 OK? 431 00:18:46,920 --> 00:18:48,910 So it matters. 432 00:18:48,910 --> 00:18:53,190 So the speed of this dot estimation task at kindergarten 433 00:18:53,190 --> 00:18:56,730 is not associated with later abilities of other kinds, 434 00:18:56,730 --> 00:18:59,760 like Raven matrices, which is one of the standard measures 435 00:18:59,760 --> 00:19:01,290 in an IQ test, right? 436 00:19:01,290 --> 00:19:05,970 It's a nonverbal and non-number kind of task. 437 00:19:05,970 --> 00:19:08,640 Or ability to name digits or letters or other things 438 00:19:08,640 --> 00:19:14,670 that you can test kids on in however old 439 00:19:14,670 --> 00:19:16,140 they are, nine years. 440 00:19:16,140 --> 00:19:16,860 OK? 441 00:19:16,860 --> 00:19:20,730 So it's specifically predictive of later arithmetic ability. 442 00:19:20,730 --> 00:19:21,900 Everybody with me? 443 00:19:21,900 --> 00:19:23,490 So it matters. 444 00:19:23,490 --> 00:19:28,230 All right, and that suggests that there would be ways 445 00:19:28,230 --> 00:19:30,000 to intervene in dyscalculia. 446 00:19:30,000 --> 00:19:31,890 Potentially, you could catch the kids early 447 00:19:31,890 --> 00:19:33,420 who are destined to have a hard time 448 00:19:33,420 --> 00:19:35,580 and maybe figure out what you could do about it. 449 00:19:35,580 --> 00:19:37,680 And there are efforts underway to do that. 450 00:19:37,680 --> 00:19:39,280 OK? 451 00:19:39,280 --> 00:19:39,780 OK. 452 00:19:42,830 --> 00:19:45,980 OK, so I'm going to show you. 453 00:19:45,980 --> 00:19:48,418 We're exploring these various number abilities. 454 00:19:48,418 --> 00:19:50,210 I'm going to show you something interesting 455 00:19:50,210 --> 00:19:51,885 about symbolic numbers. 456 00:19:51,885 --> 00:19:54,260 So far, we've been telling you about nonsymbolic numbers. 457 00:19:54,260 --> 00:19:55,823 That means just dot arrays. 458 00:19:55,823 --> 00:19:57,740 Now we're going to deal with symbolic numbers. 459 00:19:57,740 --> 00:19:59,810 I'm going to flash up a bunch of numbers. 460 00:19:59,810 --> 00:20:03,230 And you're just going to say bigger if it's bigger than 65 461 00:20:03,230 --> 00:20:06,470 or smaller if it's smaller than 65. 462 00:20:06,470 --> 00:20:07,260 Really easy. 463 00:20:07,260 --> 00:20:09,260 But you're going to shout it out loud and clear. 464 00:20:09,260 --> 00:20:10,460 Ready? 465 00:20:10,460 --> 00:20:12,124 Here we go. 466 00:20:12,124 --> 00:20:13,070 AUDIENCE: Smaller. 467 00:20:13,070 --> 00:20:13,994 NANCY KANWISHER: Good. 468 00:20:13,994 --> 00:20:14,702 AUDIENCE: Bigger. 469 00:20:14,702 --> 00:20:15,684 NANCY KANWISHER: Good. 470 00:20:15,684 --> 00:20:17,650 AUDIENCE: Smaller. 471 00:20:17,650 --> 00:20:19,540 Bigger. 472 00:20:19,540 --> 00:20:21,330 Smaller. 473 00:20:21,330 --> 00:20:23,320 Bigger. 474 00:20:23,320 --> 00:20:25,520 Smaller. 475 00:20:25,520 --> 00:20:27,083 Bigger. 476 00:20:27,083 --> 00:20:29,500 NANCY KANWISHER: OK, did you guys see what happened there? 477 00:20:29,500 --> 00:20:30,960 Did you feel what happened? 478 00:20:30,960 --> 00:20:34,900 When the numbers get closer to 65, you're slower. 479 00:20:34,900 --> 00:20:37,570 Now you think about it, why the hell is that, right? 480 00:20:37,570 --> 00:20:39,310 If you run this in Matlab, it's not 481 00:20:39,310 --> 00:20:43,720 going to take longer to tell you that 63 is smaller than 65 482 00:20:43,720 --> 00:20:48,100 than it takes to tell you that eight is smaller than 65, 483 00:20:48,100 --> 00:20:48,610 right? 484 00:20:48,610 --> 00:20:49,378 I assume. 485 00:20:49,378 --> 00:20:50,170 I haven't tried it. 486 00:20:50,170 --> 00:20:52,100 But I doubt it. 487 00:20:52,100 --> 00:20:53,720 So what does that mean? 488 00:20:53,720 --> 00:20:55,450 That means that even when you are dealing 489 00:20:55,450 --> 00:20:57,880 with symbolic numbers, numbers that you 490 00:20:57,880 --> 00:21:00,550 have this whole elaborate edifice you've been trained 491 00:21:00,550 --> 00:21:05,470 on how to operate with these guys, especially you guys, 492 00:21:05,470 --> 00:21:09,130 you are still invoking some kind of notion 493 00:21:09,130 --> 00:21:10,690 of the continuous quantity. 494 00:21:10,690 --> 00:21:12,940 You haven't totally left that idea behind 495 00:21:12,940 --> 00:21:15,770 and moved off into some abstract space. 496 00:21:15,770 --> 00:21:20,410 You're still, even in doing this very literal, exact 497 00:21:20,410 --> 00:21:25,050 symbolic number task, you find it easier 498 00:21:25,050 --> 00:21:27,550 when the numbers are farther apart than when they're closer. 499 00:21:27,550 --> 00:21:28,663 Yeah, Talia? 500 00:21:28,663 --> 00:21:31,790 AUDIENCE: Could it be because of the number you chose? 501 00:21:31,790 --> 00:21:35,530 So if you chose the numbers 60, let's say, 502 00:21:35,530 --> 00:21:37,840 I feel like we read left to right. 503 00:21:37,840 --> 00:21:39,880 And they maybe have a good concept 504 00:21:39,880 --> 00:21:41,660 for the number of digits that we see. 505 00:21:41,660 --> 00:21:44,590 So when we see a number like 62, we 506 00:21:44,590 --> 00:21:47,195 have to read both the digits instead of just the one. 507 00:21:47,195 --> 00:21:48,820 NANCY KANWISHER: Yeah, but all the ones 508 00:21:48,820 --> 00:21:50,895 I showed were at least two digits. 509 00:21:50,895 --> 00:21:51,520 AUDIENCE: Yeah. 510 00:21:51,520 --> 00:21:56,200 But when you read, like when you see a number like 25, 511 00:21:56,200 --> 00:21:57,580 you see the two. 512 00:21:57,580 --> 00:21:59,440 And then you automatically like know that. 513 00:21:59,440 --> 00:22:00,815 NANCY KANWISHER: OK, fair enough. 514 00:22:00,815 --> 00:22:03,040 OK, that's a good counter explanation. 515 00:22:03,040 --> 00:22:07,150 But you guys were slow even with 58. 516 00:22:07,150 --> 00:22:08,590 I think, right? 517 00:22:08,590 --> 00:22:09,730 We could test that. 518 00:22:09,730 --> 00:22:12,147 I'm pretty sure all this has been tested pretty carefully. 519 00:22:12,147 --> 00:22:15,310 I don't know this literature totally thoroughly. 520 00:22:15,310 --> 00:22:17,320 But I doubt-- it's a good alternative account. 521 00:22:17,320 --> 00:22:18,612 And there might be some effect. 522 00:22:18,612 --> 00:22:19,450 But I think it's-- 523 00:22:19,450 --> 00:22:22,060 oh, in fact, in fact, actually, there is, yeah, 524 00:22:22,060 --> 00:22:23,680 I have data coming up next. 525 00:22:23,680 --> 00:22:24,550 But right. 526 00:22:24,550 --> 00:22:25,270 Blah, blah. 527 00:22:25,270 --> 00:22:27,250 OK, here's the data. 528 00:22:27,250 --> 00:22:28,720 OK? 529 00:22:28,720 --> 00:22:30,640 It's pretty continuous. 530 00:22:30,640 --> 00:22:34,660 So I think your good, plausible alternative doesn't seem 531 00:22:34,660 --> 00:22:36,230 to capture very much of it. 532 00:22:36,230 --> 00:22:36,730 OK? 533 00:22:39,250 --> 00:22:41,000 So yeah, this is what you guys just did. 534 00:22:41,000 --> 00:22:43,583 And does everybody get how this kind of reveals that even when 535 00:22:43,583 --> 00:22:45,370 you think you're doing this kind of more 536 00:22:45,370 --> 00:22:48,010 symbolic abstract thing, you're still 537 00:22:48,010 --> 00:22:51,280 tapping into some kind of continuous notion? 538 00:22:51,280 --> 00:22:52,120 Yeah? 539 00:22:52,120 --> 00:22:53,320 OK. 540 00:22:53,320 --> 00:22:55,630 So that says not only does your ability 541 00:22:55,630 --> 00:22:58,390 to do that in kindergarten predict your ability 542 00:22:58,390 --> 00:23:01,900 to do arithmetic later, it says, even now as highly trained 543 00:23:01,900 --> 00:23:04,570 MIT students who do all kinds of much more 544 00:23:04,570 --> 00:23:06,280 sophisticated math than this, you're 545 00:23:06,280 --> 00:23:09,190 still invoking that same kind of continuous sense 546 00:23:09,190 --> 00:23:11,560 of approximate number or something like it. 547 00:23:11,560 --> 00:23:13,550 OK. 548 00:23:13,550 --> 00:23:15,147 All right, so where have we gotten? 549 00:23:15,147 --> 00:23:16,730 We started with this checklist of what 550 00:23:16,730 --> 00:23:18,200 number sense might mean. 551 00:23:18,200 --> 00:23:22,010 And I've argued that you adults can represent 552 00:23:22,010 --> 00:23:25,640 large numerical magnitudes without verbal counting, 553 00:23:25,640 --> 00:23:27,140 that these numbers are approximate, 554 00:23:27,140 --> 00:23:28,970 and that your ability to discriminate them 555 00:23:28,970 --> 00:23:32,220 depends on the ratio, not the difference. 556 00:23:32,220 --> 00:23:35,540 And I've sort of loosely told you 557 00:23:35,540 --> 00:23:37,040 that these experiments are generally 558 00:23:37,040 --> 00:23:40,610 done unconfounded from things like area 559 00:23:40,610 --> 00:23:43,850 and that they refer to the discrete number. 560 00:23:43,850 --> 00:23:45,860 OK, what about these other questions here? 561 00:23:45,860 --> 00:23:49,160 I haven't really shown you how abstract they are 562 00:23:49,160 --> 00:23:52,610 or whether you can actually use them in arithmetic operations. 563 00:23:52,610 --> 00:23:54,410 OK, so how would we tell that? 564 00:23:54,410 --> 00:23:56,690 Well, here's an experiment that we did way back. 565 00:23:56,690 --> 00:23:58,460 We did the very same task I did on you 566 00:23:58,460 --> 00:24:01,730 guys before, which has more, except the first thing was 567 00:24:01,730 --> 00:24:02,750 an array of dots. 568 00:24:02,750 --> 00:24:06,200 And the second thing was a series of tones. 569 00:24:06,200 --> 00:24:07,220 OK? 570 00:24:07,220 --> 00:24:10,090 Series of tones presented faster than you could count. 571 00:24:10,090 --> 00:24:13,070 Beep, beep, beep, beep, beep, like that, right? 572 00:24:13,070 --> 00:24:14,090 OK. 573 00:24:14,090 --> 00:24:15,800 And so you might think that if people 574 00:24:15,800 --> 00:24:18,260 are doing some literal perceptual thing that this 575 00:24:18,260 --> 00:24:21,980 would be just like freaking impossible, right? 576 00:24:21,980 --> 00:24:23,240 But it's not. 577 00:24:23,240 --> 00:24:25,280 Accuracy is just about the same, maybe 578 00:24:25,280 --> 00:24:27,230 a hair lower, but almost the same 579 00:24:27,230 --> 00:24:29,960 with the cross-modal comparison of which has more 580 00:24:29,960 --> 00:24:33,380 than with the within modality one, visual dots to dots 581 00:24:33,380 --> 00:24:34,443 or tones to tones. 582 00:24:34,443 --> 00:24:36,110 This is dots to dots and tones to tones. 583 00:24:36,110 --> 00:24:37,596 And that's across. 584 00:24:37,596 --> 00:24:39,215 It's a little bit surprising. 585 00:24:39,215 --> 00:24:41,340 So that shows you that whatever you're tapping into 586 00:24:41,340 --> 00:24:43,020 is a pretty abstract representation. 587 00:24:43,020 --> 00:24:44,370 It's not tied to vision. 588 00:24:44,370 --> 00:24:45,840 It's not tied to hearing. 589 00:24:45,840 --> 00:24:47,850 And it also completely eliminates 590 00:24:47,850 --> 00:24:50,100 worries about density or area or stuff 591 00:24:50,100 --> 00:24:52,140 like that because that doesn't work here at all. 592 00:24:52,140 --> 00:24:53,140 OK? 593 00:24:53,140 --> 00:24:53,640 All right. 594 00:24:56,220 --> 00:24:58,750 OK, can you do operations on these? 595 00:24:58,750 --> 00:24:59,250 Sure. 596 00:24:59,250 --> 00:25:00,000 Why not? 597 00:25:00,000 --> 00:25:02,820 You can give people a dot array and a dot array 598 00:25:02,820 --> 00:25:05,400 and then tell them to add and ask whether the sum of those 599 00:25:05,400 --> 00:25:07,450 is greater or less than that. 600 00:25:07,450 --> 00:25:10,450 Let's try it. 601 00:25:10,450 --> 00:25:11,080 OK, here we go. 602 00:25:11,080 --> 00:25:12,870 Everyone ready? 603 00:25:12,870 --> 00:25:19,210 Consider, is the sum of this plus this greater or less 604 00:25:19,210 --> 00:25:21,660 than this. 605 00:25:21,660 --> 00:25:22,410 AUDIENCE: Greater. 606 00:25:22,410 --> 00:25:24,000 NANCY KANWISHER: Yeah. 607 00:25:24,000 --> 00:25:24,960 OK? 608 00:25:24,960 --> 00:25:27,060 And I really didn't leave you time to count. 609 00:25:27,060 --> 00:25:28,830 And so whatever you were doing in adding, 610 00:25:28,830 --> 00:25:30,930 you weren't adding symbolic numbers. 611 00:25:30,930 --> 00:25:33,240 You were adding these approximate amounts. 612 00:25:33,240 --> 00:25:33,900 OK? 613 00:25:33,900 --> 00:25:36,550 Well done. 614 00:25:36,550 --> 00:25:39,880 And then we could go crazy and do it across modalities. 615 00:25:39,880 --> 00:25:42,160 I'm going to ask you to add dots to tones 616 00:25:42,160 --> 00:25:45,130 and ask whether the sum is greater or less than that. 617 00:25:45,130 --> 00:25:46,270 We won't do it. 618 00:25:46,270 --> 00:25:50,760 But it turns out, people are just as good at that. 619 00:25:50,760 --> 00:25:52,560 Amazing, huh? 620 00:25:52,560 --> 00:25:54,240 So where has this gotten us? 621 00:25:54,240 --> 00:25:56,820 This told us that whatever this approximate number sense 622 00:25:56,820 --> 00:26:00,060 that we all have, it's damned abstract. 623 00:26:00,060 --> 00:26:02,610 You can compare it across sensory modalities pretty much 624 00:26:02,610 --> 00:26:03,990 as well as within. 625 00:26:03,990 --> 00:26:05,820 And you can perform operations with it. 626 00:26:05,820 --> 00:26:06,720 You can do addition. 627 00:26:06,720 --> 00:26:09,690 And you can also do subtraction just as straightforwardly. 628 00:26:09,690 --> 00:26:11,330 OK? 629 00:26:11,330 --> 00:26:14,120 So that's pretty cool. 630 00:26:14,120 --> 00:26:18,080 But in all of these studies and the demos with you guys, 631 00:26:18,080 --> 00:26:20,900 these are done on people with years and years of training 632 00:26:20,900 --> 00:26:22,980 in arithmetic. 633 00:26:22,980 --> 00:26:25,460 And so we really want to know, are these things-- 634 00:26:25,460 --> 00:26:27,710 is any aspect of this system innate? 635 00:26:27,710 --> 00:26:29,450 Is it present in very young infants? 636 00:26:29,450 --> 00:26:32,420 And to what extent do animals have these abilities? 637 00:26:32,420 --> 00:26:33,268 OK? 638 00:26:33,268 --> 00:26:35,060 Well, how would we find out whether they're 639 00:26:35,060 --> 00:26:37,230 present in infants? 640 00:26:37,230 --> 00:26:38,480 Well, there's a bunch of ways. 641 00:26:38,480 --> 00:26:41,300 But looking direction and looking time 642 00:26:41,300 --> 00:26:44,180 are the key cues you have with newborn infants. 643 00:26:44,180 --> 00:26:49,160 And so here's a study that was done on four-day-old infants. 644 00:26:49,160 --> 00:26:52,920 And what they did was they presented-- 645 00:26:52,920 --> 00:26:56,090 they had a familiarization phase. 646 00:26:56,090 --> 00:26:58,130 This is done cross modally. 647 00:26:58,130 --> 00:26:58,730 OK? 648 00:26:58,730 --> 00:27:02,780 So they present either sets of 12 sounds, 649 00:27:02,780 --> 00:27:07,490 to, to, to, to, 12, right, of those, or ra, ra, ra, ra, 650 00:27:07,490 --> 00:27:09,590 present a bunch of those to infants. 651 00:27:09,590 --> 00:27:13,580 Or they present sets of four taking the same total duration, 652 00:27:13,580 --> 00:27:15,728 to, to. 653 00:27:15,728 --> 00:27:17,270 It's just a coincidence that it's to. 654 00:27:17,270 --> 00:27:19,070 This is, I think, done in French. 655 00:27:19,070 --> 00:27:23,960 So anyway, the infants won't be confused by the sound to. 656 00:27:23,960 --> 00:27:30,350 So during that, you then show the infants these arrays. 657 00:27:30,350 --> 00:27:34,230 And you ask what they look at more. 658 00:27:34,230 --> 00:27:35,840 OK? 659 00:27:35,840 --> 00:27:37,052 They're not told the task. 660 00:27:37,052 --> 00:27:38,510 There's no way to tell them a task. 661 00:27:38,510 --> 00:27:40,520 It's just something they do. 662 00:27:40,520 --> 00:27:46,250 And what you find is, in the four versus 12 case 663 00:27:46,250 --> 00:27:49,100 like here, that's four versus 12, 664 00:27:49,100 --> 00:27:53,700 the infants look more at the congruent number then 665 00:27:53,700 --> 00:27:56,110 the incongruent number. 666 00:27:56,110 --> 00:27:56,610 OK? 667 00:27:56,610 --> 00:27:59,010 So again, they're comparing across modality. 668 00:27:59,010 --> 00:28:02,670 They're hearing some number of syllables. 669 00:28:02,670 --> 00:28:06,060 And they're selectively looking at the corresponding number 670 00:28:06,060 --> 00:28:08,400 of visual forms. 671 00:28:08,400 --> 00:28:09,210 No instruction. 672 00:28:09,210 --> 00:28:09,960 No nothing. 673 00:28:09,960 --> 00:28:11,070 Four days old. 674 00:28:11,070 --> 00:28:12,180 Amazing. 675 00:28:12,180 --> 00:28:13,140 OK? 676 00:28:13,140 --> 00:28:16,320 So they can do that if the comparison is four versus 12. 677 00:28:16,320 --> 00:28:19,650 They can do it if it's six versus 18. 678 00:28:19,650 --> 00:28:21,330 But they kind of can't do it very well. 679 00:28:21,330 --> 00:28:23,340 I mean, it's significant, but it's not very good 680 00:28:23,340 --> 00:28:25,480 if it's four versus eight. 681 00:28:25,480 --> 00:28:25,980 OK? 682 00:28:25,980 --> 00:28:28,350 So they have some sense of number. 683 00:28:28,350 --> 00:28:29,920 But it's very approximate. 684 00:28:29,920 --> 00:28:30,420 Yeah? 685 00:28:30,420 --> 00:28:31,020 AUDIENCE: Did you say they looked 686 00:28:31,020 --> 00:28:32,860 at the one that matches the number, 687 00:28:32,860 --> 00:28:34,568 or they hear the sound that goes with it? 688 00:28:34,568 --> 00:28:35,610 NANCY KANWISHER: Matches. 689 00:28:35,610 --> 00:28:37,260 That's congruent means match. 690 00:28:37,260 --> 00:28:40,450 Looking time on congruent versus incongruent. 691 00:28:40,450 --> 00:28:42,471 AUDIENCE: Isn't that kind of different from-- 692 00:28:42,471 --> 00:28:43,015 NANCY KANWISHER: From adaptation. 693 00:28:43,015 --> 00:28:43,690 AUDIENCE: Yeah. 694 00:28:43,690 --> 00:28:44,590 NANCY KANWISHER: It is. 695 00:28:44,590 --> 00:28:46,257 It is totally different from adaptation. 696 00:28:46,257 --> 00:28:49,870 And herein lies a classic annoyance 697 00:28:49,870 --> 00:28:51,580 for developmental psychologists. 698 00:28:51,580 --> 00:28:54,250 Because sometimes kids match. 699 00:28:54,250 --> 00:28:56,530 And sometimes they show adaptation. 700 00:28:56,530 --> 00:28:59,920 And you kind of don't know. 701 00:28:59,920 --> 00:29:03,940 Sometimes you don't know which way it's going to go. 702 00:29:03,940 --> 00:29:04,575 I don't know. 703 00:29:04,575 --> 00:29:06,700 Heather, do we have any insights about how you know 704 00:29:06,700 --> 00:29:07,825 which way it's going to go? 705 00:29:07,825 --> 00:29:11,740 Or you just try and experiment and you find out and yeah? 706 00:29:11,740 --> 00:29:12,280 Yeah. 707 00:29:12,280 --> 00:29:12,780 Yeah. 708 00:29:12,780 --> 00:29:14,458 It does mean you have to be careful. 709 00:29:14,458 --> 00:29:16,000 Because if you run a whole experiment 710 00:29:16,000 --> 00:29:17,560 on a smallish number of infants-- 711 00:29:17,560 --> 00:29:19,720 and it's usually hard to get enough because people 712 00:29:19,720 --> 00:29:21,310 have to drive in with their kids. 713 00:29:21,310 --> 00:29:22,610 And this, how do you find them? 714 00:29:22,610 --> 00:29:24,850 And there's other developmental labs who have all the kids. 715 00:29:24,850 --> 00:29:26,808 And it's like you're always running experiments 716 00:29:26,808 --> 00:29:29,470 with barely enough kids, right? 717 00:29:29,470 --> 00:29:31,330 And so that means there's a problem here. 718 00:29:31,330 --> 00:29:34,750 Because if you would take the result in either direction, 719 00:29:34,750 --> 00:29:36,460 that's a statistical problem. 720 00:29:36,460 --> 00:29:38,530 You gave yourself two shots at it, right? 721 00:29:38,530 --> 00:29:42,490 And so you have to statistically discount your finding 722 00:29:42,490 --> 00:29:44,380 because it could have gone either way. 723 00:29:44,380 --> 00:29:45,863 That is if your prior hypothesis is 724 00:29:45,863 --> 00:29:48,280 it has to go in one direction, you're on stronger footing. 725 00:29:48,280 --> 00:29:51,580 But you just suck it up and run a few more kids. 726 00:29:51,580 --> 00:29:52,990 Yeah? 727 00:29:52,990 --> 00:29:56,230 OK, so good. 728 00:29:56,230 --> 00:30:00,595 So this also shows that ratio dependence, right? 729 00:30:00,595 --> 00:30:02,470 They're better at it with the big differences 730 00:30:02,470 --> 00:30:03,890 than the small differences. 731 00:30:03,890 --> 00:30:04,390 OK? 732 00:30:07,190 --> 00:30:11,240 OK, so that's infants having this very, very early, at least 733 00:30:11,240 --> 00:30:13,740 in very crude form. 734 00:30:13,740 --> 00:30:15,170 What about animals? 735 00:30:15,170 --> 00:30:18,710 OK, so let's meet Mercury the macaw. 736 00:30:18,710 --> 00:30:20,105 Here's Mercury the macaw. 737 00:30:20,105 --> 00:30:20,772 [VIDEO PLAYBACK] 738 00:30:20,772 --> 00:30:22,550 - To a human, the order of the symbols 739 00:30:22,550 --> 00:30:24,487 shown on the above screen are obvious. 740 00:30:24,487 --> 00:30:26,945 We have all learned from a young age which of these symbols 741 00:30:26,945 --> 00:30:27,170 represented-- 742 00:30:27,170 --> 00:30:27,290 [END PLAYBACK] 743 00:30:27,290 --> 00:30:27,980 NANCY KANWISHER: Oh, what a good birdie. 744 00:30:27,980 --> 00:30:28,647 [VIDEO PLAYBACK] 745 00:30:28,647 --> 00:30:31,280 - --the lowest number and which the highest. 746 00:30:31,280 --> 00:30:35,360 However, for Mercury, the blue-headed macaw we see here, 747 00:30:35,360 --> 00:30:37,430 he has had to learn by trial and error 748 00:30:37,430 --> 00:30:39,680 the specific order to press these symbols to get 749 00:30:39,680 --> 00:30:41,240 a piece of food. 750 00:30:41,240 --> 00:30:44,480 It took him quite a long time. 751 00:30:44,480 --> 00:30:47,810 Mercury's brother Mars can do a bit better than that. 752 00:30:47,810 --> 00:30:50,180 He has begun to learn the more general concept. 753 00:30:50,180 --> 00:30:53,640 That is the symbols will always have an order. 754 00:30:53,640 --> 00:30:55,640 So when presented with a new list, 755 00:30:55,640 --> 00:30:58,240 he was able to rapidly decipher the order 756 00:30:58,240 --> 00:31:00,710 of new symbols, in this case kingfisher, 757 00:31:00,710 --> 00:31:04,040 warhead, hawk, hummingbird. 758 00:31:04,040 --> 00:31:05,990 Pressing randomly on the screen would 759 00:31:05,990 --> 00:31:08,870 have led to him receiving the correct answer less than 1% 760 00:31:08,870 --> 00:31:09,890 of the time. 761 00:31:09,890 --> 00:31:12,050 He's clearly doing better than that. 762 00:31:12,050 --> 00:31:15,080 This is interesting, as it shows the very basic aspects 763 00:31:15,080 --> 00:31:17,390 of cognition related to numbers are 764 00:31:17,390 --> 00:31:19,610 present in an animal that is very distantly 765 00:31:19,610 --> 00:31:21,694 related to humans. 766 00:31:21,694 --> 00:31:22,580 [END PLAYBACK] 767 00:31:22,580 --> 00:31:23,780 NANCY KANWISHER: OK, mostly, I just showed 768 00:31:23,780 --> 00:31:24,738 that because it's cute. 769 00:31:24,738 --> 00:31:26,780 But it's impressive ordering. 770 00:31:26,780 --> 00:31:27,290 OK? 771 00:31:27,290 --> 00:31:29,120 Still, he's kind of slow. 772 00:31:29,120 --> 00:31:32,180 I think it only goes up to four things. 773 00:31:32,180 --> 00:31:35,930 OK, so now we're going to meet the chimp Ayumu, 774 00:31:35,930 --> 00:31:38,600 who lives in Kyoto and who's the son of a very 775 00:31:38,600 --> 00:31:42,020 famous chimp named Ai, who was like a number wiz. 776 00:31:42,020 --> 00:31:44,105 But anyway, here's Ayumu. 777 00:31:44,105 --> 00:31:46,750 [VIDEO PLAYBACK] 778 00:31:53,347 --> 00:31:53,930 [END PLAYBACK] 779 00:31:53,930 --> 00:31:54,930 NANCY KANWISHER: I know. 780 00:31:54,930 --> 00:31:56,420 I can only catch the first three. 781 00:31:56,420 --> 00:31:58,628 And then it's like I can't even tell if he's correct, 782 00:31:58,628 --> 00:32:01,460 except from the tone. 783 00:32:01,460 --> 00:32:02,120 Pretty good. 784 00:32:02,120 --> 00:32:05,578 [VIDEO PLAYBACK] 785 00:32:13,790 --> 00:32:14,470 [END PLAYBACK] 786 00:32:14,470 --> 00:32:15,928 NANCY KANWISHER: Oh, got one wrong. 787 00:32:20,818 --> 00:32:22,110 Anyway, mostly gets them right. 788 00:32:22,110 --> 00:32:24,210 Pretty impressive, huh? 789 00:32:24,210 --> 00:32:25,680 OK, so that's cool. 790 00:32:25,680 --> 00:32:27,720 And order is clearly relevant. 791 00:32:27,720 --> 00:32:28,740 It's part of the space. 792 00:32:28,740 --> 00:32:32,190 But it's not the same as quantity or number, right? 793 00:32:32,190 --> 00:32:35,700 OK, so now we're going to skip to the honeybee, 794 00:32:35,700 --> 00:32:37,650 just for kicks because this paper just 795 00:32:37,650 --> 00:32:38,890 came out a month ago. 796 00:32:38,890 --> 00:32:41,320 And I think it's awesome. 797 00:32:41,320 --> 00:32:43,837 Honeybees have 1 million neurons. 798 00:32:43,837 --> 00:32:45,670 And if you're impressed, don't be impressed. 799 00:32:45,670 --> 00:32:48,600 Remember like a mouse has 100 million. 800 00:32:48,600 --> 00:32:51,370 And we have 100 billion. 801 00:32:51,370 --> 00:32:51,960 OK? 802 00:32:51,960 --> 00:32:53,460 Six orders of magnitude. 803 00:32:53,460 --> 00:32:56,670 OK, so 1 million is like not-- no. 804 00:32:56,670 --> 00:32:57,910 eight orders of magnitude. 805 00:32:57,910 --> 00:33:01,020 So 1 million is not that many, right? 806 00:33:01,020 --> 00:33:03,000 OK, and further, these guys branched off 807 00:33:03,000 --> 00:33:05,800 from us, evolutionarily, a very long time ago, 808 00:33:05,800 --> 00:33:07,240 600 million years ago. 809 00:33:07,240 --> 00:33:09,120 So they're tiny little guys, not very 810 00:33:09,120 --> 00:33:11,220 many neurons, totally different kind of thing. 811 00:33:11,220 --> 00:33:14,430 Who would think they have any kind of numerical abilities? 812 00:33:14,430 --> 00:33:16,360 Of course, they wouldn't, right? 813 00:33:16,360 --> 00:33:20,130 Oh, and yet, they can do arithmetic. 814 00:33:20,130 --> 00:33:22,750 OK, so here's the design. 815 00:33:22,750 --> 00:33:24,840 So here's what these guys did, this wonderful lab 816 00:33:24,840 --> 00:33:25,440 in Australia. 817 00:33:25,440 --> 00:33:26,220 I love this stuff. 818 00:33:26,220 --> 00:33:28,680 OK, so they trained these honeybees. 819 00:33:28,680 --> 00:33:31,710 This was a chamber like this. 820 00:33:31,710 --> 00:33:33,390 Honeybees fly into the chamber. 821 00:33:33,390 --> 00:33:36,360 And they see a number in a color right here. 822 00:33:36,360 --> 00:33:37,090 It's blue. 823 00:33:37,090 --> 00:33:37,920 And it's two. 824 00:33:37,920 --> 00:33:38,580 OK? 825 00:33:38,580 --> 00:33:40,410 And then there's a little entry hole. 826 00:33:40,410 --> 00:33:42,900 And they can choose to play or not play. 827 00:33:42,900 --> 00:33:46,050 If they go into the chamber, then they're 828 00:33:46,050 --> 00:33:47,940 in this interior space, where they 829 00:33:47,940 --> 00:33:51,480 get to make a choice between that pattern and that pattern. 830 00:33:51,480 --> 00:33:52,380 OK? 831 00:33:52,380 --> 00:33:55,360 And there's a little pole underneath each pattern. 832 00:33:55,360 --> 00:33:57,690 And if they light and they land on the pole, 833 00:33:57,690 --> 00:33:59,500 they can get some liquid. 834 00:33:59,500 --> 00:34:00,000 OK? 835 00:34:00,000 --> 00:34:04,500 So in the blue case, they're rewarded over trials. 836 00:34:04,500 --> 00:34:06,330 That if it's blue, that means they 837 00:34:06,330 --> 00:34:08,406 should add one to this number. 838 00:34:08,406 --> 00:34:10,239 And hence, that would be the correct answer. 839 00:34:10,239 --> 00:34:12,460 And that's the incorrect answer. 840 00:34:12,460 --> 00:34:13,180 OK? 841 00:34:13,180 --> 00:34:14,290 That would be amazing. 842 00:34:14,290 --> 00:34:15,520 Yeah? 843 00:34:15,520 --> 00:34:17,020 And if they choose the wrong number, 844 00:34:17,020 --> 00:34:19,540 they get some nasty quinine. 845 00:34:19,540 --> 00:34:20,770 OK? 846 00:34:20,770 --> 00:34:26,020 All right, in contrast, if the shape out front is yellow, 847 00:34:26,020 --> 00:34:28,090 then they have to subtract. 848 00:34:28,090 --> 00:34:30,310 So that means they have to keep track of this number 849 00:34:30,310 --> 00:34:34,005 and go in there and choose that number minus one. 850 00:34:34,005 --> 00:34:35,949 All right? 851 00:34:35,949 --> 00:34:40,150 OK, so keep in mind, oh, so they balance the total surface area. 852 00:34:40,150 --> 00:34:41,650 It doesn't look like in this figure. 853 00:34:41,650 --> 00:34:43,659 But it says in the method section they did. 854 00:34:43,659 --> 00:34:45,699 I believe them. 855 00:34:45,699 --> 00:34:49,480 And further, realize that when the bee is in here, 856 00:34:49,480 --> 00:34:53,980 he has to be holding that number in memory and adding one to it 857 00:34:53,980 --> 00:34:56,960 or subtracting one to it to figure out what to choose here. 858 00:34:56,960 --> 00:34:58,600 So this is pretty sophisticated. 859 00:34:58,600 --> 00:35:01,780 It's not like they're side by side, right? 860 00:35:01,780 --> 00:35:05,740 OK, and yet, they're pretty good at it. 861 00:35:05,740 --> 00:35:07,900 Here's accuracy over training trials. 862 00:35:07,900 --> 00:35:12,320 By 100 trials, they're over 80% correct. 863 00:35:12,320 --> 00:35:14,270 Pretty amazing, isn't it? 864 00:35:14,270 --> 00:35:19,580 OK, so then, in any good animal or infant cognition study, 865 00:35:19,580 --> 00:35:21,530 you want to show whether it generalizes. 866 00:35:21,530 --> 00:35:24,740 So then they test the same ability with new numbers. 867 00:35:24,740 --> 00:35:26,120 I forget what this range was. 868 00:35:26,120 --> 00:35:28,200 But it went one to four or something like that. 869 00:35:28,200 --> 00:35:32,210 And then they go to five or six, just to generalize the numbers, 870 00:35:32,210 --> 00:35:35,330 and different shapes than were used in the training trial. 871 00:35:35,330 --> 00:35:39,080 And the accuracy is around mid 60s. 872 00:35:39,080 --> 00:35:40,438 It's not quite as good. 873 00:35:40,438 --> 00:35:41,480 But it's still very good. 874 00:35:41,480 --> 00:35:42,897 They're not being reinforced here. 875 00:35:42,897 --> 00:35:44,720 And they're still doing the task. 876 00:35:44,720 --> 00:35:46,520 Now, what are the pink and blue bars? 877 00:35:46,520 --> 00:35:50,420 OK, so you might think, well, is a bee just 878 00:35:50,420 --> 00:35:52,170 going to the one that has more or less? 879 00:35:52,170 --> 00:35:54,140 So instead of learning add one, he's 880 00:35:54,140 --> 00:35:57,200 learned go to the larger number, larger than the one 881 00:35:57,200 --> 00:35:59,060 that you saw at the entry chamber, 882 00:35:59,060 --> 00:36:00,590 or go to the smaller number. 883 00:36:00,590 --> 00:36:02,870 But no, that's not what they're doing. 884 00:36:02,870 --> 00:36:06,800 Because the pink bars show the performance 885 00:36:06,800 --> 00:36:09,980 when both of the options are in the same direction, right? 886 00:36:09,980 --> 00:36:13,160 So the thing is blue. 887 00:36:13,160 --> 00:36:15,140 So he's doing addition. 888 00:36:15,140 --> 00:36:16,250 And he sees a two. 889 00:36:16,250 --> 00:36:19,520 And he goes in, he has a choice between three or four. 890 00:36:19,520 --> 00:36:21,980 He can only do that if he knows the difference 891 00:36:21,980 --> 00:36:24,050 between adding one and just taking 892 00:36:24,050 --> 00:36:26,360 the thing that has more, right? 893 00:36:26,360 --> 00:36:30,200 And he's well above chance in the pink bars. 894 00:36:30,200 --> 00:36:32,750 OK, so he's not just saying, choose 895 00:36:32,750 --> 00:36:34,790 the one that has more or the one that has less. 896 00:36:34,790 --> 00:36:37,650 He's adding one, pretty accurately, 897 00:36:37,650 --> 00:36:38,870 I mean sort of accurately. 898 00:36:38,870 --> 00:36:40,510 Better than chance. 899 00:36:40,510 --> 00:36:42,540 OK? 900 00:36:42,540 --> 00:36:45,660 All right, now that's pretty cool. 901 00:36:45,660 --> 00:36:49,950 But adding one, subtracting one, it's cool. 902 00:36:49,950 --> 00:36:54,060 But do they really have abstract concepts? 903 00:36:54,060 --> 00:36:58,470 Do they understand the concept of zero? 904 00:36:58,470 --> 00:37:00,428 OK, so paper was published last year 905 00:37:00,428 --> 00:37:02,220 arguing that they have the concept of zero. 906 00:37:02,220 --> 00:37:03,840 Here's how it goes. 907 00:37:03,840 --> 00:37:06,390 Same lab trains them, in this case, 908 00:37:06,390 --> 00:37:08,950 just on greater than or less than. 909 00:37:08,950 --> 00:37:11,880 So the bees are given a choice like this. 910 00:37:11,880 --> 00:37:14,580 And one set of bees is trained on greater than 911 00:37:14,580 --> 00:37:15,940 and one is trained on less than. 912 00:37:15,940 --> 00:37:19,110 So this set of bees trained on greater than chooses 913 00:37:19,110 --> 00:37:22,650 this one and then this one and then this one and so on. 914 00:37:22,650 --> 00:37:24,060 OK? 915 00:37:24,060 --> 00:37:26,790 Another set of bees is trained to do the opposite. 916 00:37:26,790 --> 00:37:27,750 All right? 917 00:37:27,750 --> 00:37:29,910 OK, so that's the training phase. 918 00:37:29,910 --> 00:37:33,820 Then we want to test in a generalized situation. 919 00:37:33,820 --> 00:37:36,930 So now they're tested with different shapes 920 00:37:36,930 --> 00:37:41,547 and different numbers, so threes and fours were. 921 00:37:41,547 --> 00:37:42,630 Maybe threes weren't used. 922 00:37:42,630 --> 00:37:42,930 I forget. 923 00:37:42,930 --> 00:37:45,222 There's some numbers in here that were not used before. 924 00:37:45,222 --> 00:37:48,360 OK, so you test them with new shapes. 925 00:37:48,360 --> 00:37:52,200 And here is accuracy for less than or greater than. 926 00:37:52,200 --> 00:37:53,400 Chance is 50%. 927 00:37:53,400 --> 00:37:55,410 And they're 75%. 928 00:37:55,410 --> 00:37:56,790 Not bad. 929 00:37:56,790 --> 00:37:57,630 OK? 930 00:37:57,630 --> 00:38:00,120 So they get more than or less than. 931 00:38:00,120 --> 00:38:03,260 OK, now we want to test the generalization. 932 00:38:03,260 --> 00:38:04,220 OK, oh, yes, sorry. 933 00:38:04,220 --> 00:38:06,270 This is where they changed the range of numbers. 934 00:38:06,270 --> 00:38:11,013 So the bees had not dealt with sixes before. 935 00:38:11,013 --> 00:38:13,055 So now they still have to do greater than or less 936 00:38:13,055 --> 00:38:16,880 than with a new numerical range. 937 00:38:16,880 --> 00:38:19,430 And they're still well above chance. 938 00:38:19,430 --> 00:38:21,530 OK? 939 00:38:21,530 --> 00:38:25,330 So then finally, they test zero. 940 00:38:25,330 --> 00:38:26,350 OK? 941 00:38:26,350 --> 00:38:30,070 So the bees that have to do less than 942 00:38:30,070 --> 00:38:33,610 have to say which of those is correct, all right? 943 00:38:33,610 --> 00:38:35,800 And you can see-- 944 00:38:35,800 --> 00:38:38,980 where did it go? 945 00:38:38,980 --> 00:38:39,910 Where's the zero one? 946 00:38:39,910 --> 00:38:40,667 Right here. 947 00:38:40,667 --> 00:38:42,250 And they're well above chance for both 948 00:38:42,250 --> 00:38:44,440 less than and greater than. 949 00:38:44,440 --> 00:38:45,250 OK? 950 00:38:45,250 --> 00:38:46,960 So we could quibble about whether that's 951 00:38:46,960 --> 00:38:48,400 a concept of zero. 952 00:38:48,400 --> 00:38:50,860 But the cool thing is these bees had not been tested 953 00:38:50,860 --> 00:38:52,840 with a blank card before. 954 00:38:52,840 --> 00:38:58,510 And they spontaneously get the idea that that is less than one 955 00:38:58,510 --> 00:39:00,400 or two or three or anything else. 956 00:39:00,400 --> 00:39:01,970 Yeah? 957 00:39:01,970 --> 00:39:05,060 So arguably, they have a concept of zero 958 00:39:05,060 --> 00:39:09,390 with no training and only 100 million neurons. 959 00:39:09,390 --> 00:39:12,810 OK, so all of that is in trained animals. 960 00:39:12,810 --> 00:39:16,500 And we can see some of these kinds of abilities 961 00:39:16,500 --> 00:39:18,210 even with untrained animals. 962 00:39:18,210 --> 00:39:21,300 And I will tell you just one more animal experiment 963 00:39:21,300 --> 00:39:23,077 because it's my all-time favorite ever 964 00:39:23,077 --> 00:39:24,660 and the simplest one in the whole set. 965 00:39:24,660 --> 00:39:27,140 This was done a long time ago by Church and Meck. 966 00:39:27,140 --> 00:39:28,140 So here's what they did. 967 00:39:28,140 --> 00:39:29,310 This is done in rats. 968 00:39:29,310 --> 00:39:30,900 They have a training phase, where 969 00:39:30,900 --> 00:39:33,510 they train the rats to press the two 970 00:39:33,510 --> 00:39:37,830 lever if they see two light flashes or hear two sounds. 971 00:39:37,830 --> 00:39:40,110 And they press another lever, the four lever 972 00:39:40,110 --> 00:39:42,910 if they see four lights or hear four sounds. 973 00:39:42,910 --> 00:39:43,410 OK? 974 00:39:43,410 --> 00:39:44,957 That's kind of basic animal training. 975 00:39:44,957 --> 00:39:45,540 It's a rodent. 976 00:39:45,540 --> 00:39:46,415 They're good at this. 977 00:39:46,415 --> 00:39:47,470 No big deal. 978 00:39:47,470 --> 00:39:49,680 But then after the animals have learned this, 979 00:39:49,680 --> 00:39:53,190 they spontaneously throw, in the testing phase, 980 00:39:53,190 --> 00:39:56,720 a trial with two lights and two sounds. 981 00:39:56,720 --> 00:39:59,540 And the rats press the four lever, first time. 982 00:39:59,540 --> 00:40:00,290 No training. 983 00:40:00,290 --> 00:40:01,160 No nothing. 984 00:40:01,160 --> 00:40:02,600 Spontaneous addition. 985 00:40:02,600 --> 00:40:07,430 Spontaneous abstraction across tones and lights. 986 00:40:07,430 --> 00:40:10,200 Pretty awesome, huh? 987 00:40:10,200 --> 00:40:13,110 So it's not just that you can reveal these abilities 988 00:40:13,110 --> 00:40:15,580 with elaborate training. 989 00:40:15,580 --> 00:40:19,030 OK, so we have all of these different kinds of evidence 990 00:40:19,030 --> 00:40:20,860 of an abstract number sense. 991 00:40:20,860 --> 00:40:23,380 And they're present in newborn infants. 992 00:40:23,380 --> 00:40:25,150 And they're present in animals. 993 00:40:25,150 --> 00:40:28,210 And they just seem to be part of our basic cognitive machinery, 994 00:40:28,210 --> 00:40:31,040 machinery that we share with animals. 995 00:40:31,040 --> 00:40:34,600 So how are they implemented in the brain? 996 00:40:34,600 --> 00:40:38,440 OK, so a little neuroanatomy reminder of some basics. 997 00:40:38,440 --> 00:40:40,060 This is a weird angle of a brain. 998 00:40:40,060 --> 00:40:42,490 It's kind of like this, kind of back of the head, 999 00:40:42,490 --> 00:40:44,048 front of the head, temporal lobe, 1000 00:40:44,048 --> 00:40:45,340 frontal lobe around the corner. 1001 00:40:45,340 --> 00:40:46,900 Everybody oriented? 1002 00:40:46,900 --> 00:40:49,690 There is one of the longest sulci in the brain that 1003 00:40:49,690 --> 00:40:50,748 starts about here. 1004 00:40:50,748 --> 00:40:51,790 On me, it goes like this. 1005 00:40:51,790 --> 00:40:53,290 And it curves around like that. 1006 00:40:53,290 --> 00:40:53,915 It's back here. 1007 00:40:53,915 --> 00:40:54,700 It goes up. 1008 00:40:54,700 --> 00:40:55,930 And it curves over. 1009 00:40:55,930 --> 00:40:56,650 OK? 1010 00:40:56,650 --> 00:40:58,660 It's called the intraparietal sulcus. 1011 00:40:58,660 --> 00:41:00,310 And I mention that just because it's 1012 00:41:00,310 --> 00:41:02,050 in a lot of the number literature. 1013 00:41:02,050 --> 00:41:05,650 You saw it in the paper you guys read for last night. 1014 00:41:05,650 --> 00:41:09,130 And above it is the superior parietal lobule. 1015 00:41:09,130 --> 00:41:11,320 And below it is the inferior parietal lobule. 1016 00:41:11,320 --> 00:41:14,110 And none of that matters other than that a lot of the action 1017 00:41:14,110 --> 00:41:16,480 is in the parietal lobe, particularly up 1018 00:41:16,480 --> 00:41:18,490 here around the intraparietal sulcus. 1019 00:41:18,490 --> 00:41:19,780 OK? 1020 00:41:19,780 --> 00:41:22,780 All right, so studies that have looked at this 1021 00:41:22,780 --> 00:41:27,460 includes some classical studies of patients with brain damage 1022 00:41:27,460 --> 00:41:29,950 and something called acalculia. 1023 00:41:29,950 --> 00:41:32,650 That means loss of ability to calculate. 1024 00:41:32,650 --> 00:41:33,430 OK? 1025 00:41:33,430 --> 00:41:38,140 And so there's two basic kinds of acalculia 1026 00:41:38,140 --> 00:41:40,330 that are really interestingly different. 1027 00:41:40,330 --> 00:41:43,690 So there's one acalculic patient who 1028 00:41:43,690 --> 00:41:46,210 has left parietal lobe damage, that same region I just 1029 00:41:46,210 --> 00:41:47,990 talked about. 1030 00:41:47,990 --> 00:41:49,970 And this person is bad at approximation. 1031 00:41:49,970 --> 00:41:53,980 So the kinds of dot array tasks that I gave you guys, 1032 00:41:53,980 --> 00:41:55,900 this guy, after brain damage right here, 1033 00:41:55,900 --> 00:41:58,150 is really bad at that kind of stuff. 1034 00:41:58,150 --> 00:42:01,960 And interestingly, he's more impaired on subtraction 1035 00:42:01,960 --> 00:42:03,830 than multiplication. 1036 00:42:03,830 --> 00:42:08,620 So for example, hes worse at, what is seven minus five 1037 00:42:08,620 --> 00:42:11,745 than what is seven times five? 1038 00:42:11,745 --> 00:42:13,120 So think about that for a moment. 1039 00:42:13,120 --> 00:42:15,100 And think about what that might mean, 1040 00:42:15,100 --> 00:42:19,160 especially in light of another acalculic patient who has 1041 00:42:19,160 --> 00:42:20,410 a very different presentation. 1042 00:42:20,410 --> 00:42:22,720 He's got left temporal damage. 1043 00:42:22,720 --> 00:42:24,400 His approximation is fine. 1044 00:42:24,400 --> 00:42:27,310 So all those kind of dot array kind of tasks and tone tasks 1045 00:42:27,310 --> 00:42:30,250 that I told you about, he's good at. 1046 00:42:30,250 --> 00:42:32,110 This guy shows the opposite. 1047 00:42:32,110 --> 00:42:33,850 He's more impaired at multiplication 1048 00:42:33,850 --> 00:42:36,910 than subtraction. 1049 00:42:36,910 --> 00:42:39,720 So do you guys have any-- oh, so first of all, 1050 00:42:39,720 --> 00:42:44,130 you put these two patients together, and what do you have? 1051 00:42:44,130 --> 00:42:45,980 AUDIENCE: Double dissociation. 1052 00:42:45,980 --> 00:42:46,240 NANCY KANWISHER: Yeah? 1053 00:42:46,240 --> 00:42:46,480 What? 1054 00:42:46,480 --> 00:42:47,140 AUDIENCE: Double dissociation. 1055 00:42:47,140 --> 00:42:48,190 NANCY KANWISHER: Double dissociation. 1056 00:42:48,190 --> 00:42:48,690 Right. 1057 00:42:48,690 --> 00:42:52,060 Two patients with opposite patterns of deficit, right? 1058 00:42:52,060 --> 00:42:54,523 If we just had one, then we could maybe tell a story. 1059 00:42:54,523 --> 00:42:55,690 But it wouldn't really know. 1060 00:42:55,690 --> 00:42:58,013 But we have two, and they have opposite patterns. 1061 00:42:58,013 --> 00:43:00,430 And now that really kind of constrains the interpretation. 1062 00:43:00,430 --> 00:43:00,930 David. 1063 00:43:00,930 --> 00:43:03,700 AUDIENCE: Can the first person add fine? 1064 00:43:03,700 --> 00:43:05,440 NANCY KANWISHER: Good question. 1065 00:43:05,440 --> 00:43:06,908 He's not very good at adding. 1066 00:43:06,908 --> 00:43:07,450 AUDIENCE: Oh. 1067 00:43:10,207 --> 00:43:11,290 NANCY KANWISHER: Thoughts? 1068 00:43:11,290 --> 00:43:13,050 What do you think it means? 1069 00:43:13,050 --> 00:43:18,430 AUDIENCE: It might mean that the addition and subtraction 1070 00:43:18,430 --> 00:43:22,345 [INAUDIBLE] use the same like-- 1071 00:43:22,345 --> 00:43:23,470 NANCY KANWISHER: Used what? 1072 00:43:23,470 --> 00:43:25,283 AUDIENCE: Like they use the same area. 1073 00:43:25,283 --> 00:43:26,200 NANCY KANWISHER: Yeah. 1074 00:43:26,200 --> 00:43:29,140 So one hypothesis is that addition and subtraction 1075 00:43:29,140 --> 00:43:31,945 are just a different beast than multiplication. 1076 00:43:31,945 --> 00:43:33,820 Different parts of the brain do those things. 1077 00:43:33,820 --> 00:43:35,260 Totally possible. 1078 00:43:35,260 --> 00:43:37,750 But there's a kind of more intuitive interpretation. 1079 00:43:37,750 --> 00:43:40,450 AUDIENCE: Well, I think people tend to memorize times tables. 1080 00:43:40,450 --> 00:43:41,680 NANCY KANWISHER: Bingo. 1081 00:43:41,680 --> 00:43:42,200 Bingo. 1082 00:43:42,200 --> 00:43:44,200 Often, like the right answer is something that's 1083 00:43:44,200 --> 00:43:45,010 like right in front of you. 1084 00:43:45,010 --> 00:43:46,885 Just think about, what is it like to do that? 1085 00:43:46,885 --> 00:43:48,400 How do you do seven times five? 1086 00:43:48,400 --> 00:43:50,560 You don't think about the meanings of the numbers. 1087 00:43:50,560 --> 00:43:52,750 You just blurt out 35. 1088 00:43:52,750 --> 00:43:54,020 Right? 1089 00:43:54,020 --> 00:43:54,520 Right? 1090 00:43:54,520 --> 00:43:57,640 It's not a very rich number task. 1091 00:43:57,640 --> 00:43:58,810 I mean, it's a number task. 1092 00:43:58,810 --> 00:44:02,680 But it's a concrete, rote, verbally memorized thing. 1093 00:44:02,680 --> 00:44:03,880 Right? 1094 00:44:03,880 --> 00:44:11,200 And so the idea is that those verbalized concrete number 1095 00:44:11,200 --> 00:44:13,960 facts are in one domain. 1096 00:44:13,960 --> 00:44:16,060 One set of brain damage would impair those. 1097 00:44:16,060 --> 00:44:17,800 And it's a different thing to impair 1098 00:44:17,800 --> 00:44:20,770 the actual representation of numerosity. 1099 00:44:20,770 --> 00:44:25,090 And the idea is that this person is 1100 00:44:25,090 --> 00:44:28,460 the one with the real damage to the approximate number system. 1101 00:44:28,460 --> 00:44:28,960 Right? 1102 00:44:28,960 --> 00:44:29,740 Yeah? 1103 00:44:29,740 --> 00:44:32,920 AUDIENCE: Does that mean that patient can be it 1104 00:44:32,920 --> 00:44:36,970 is a problem doing the seven times five normally. 1105 00:44:36,970 --> 00:44:40,730 But when they ask for summing seven for five times, 1106 00:44:40,730 --> 00:44:42,113 they're not very good. 1107 00:44:42,113 --> 00:44:43,030 NANCY KANWISHER: Yeah. 1108 00:44:43,030 --> 00:44:44,822 Well, I think the approximate number system 1109 00:44:44,822 --> 00:44:49,750 might have a tough time dealing with summing seven five times. 1110 00:44:49,750 --> 00:44:51,610 So yeah, it has limits, right? 1111 00:44:51,610 --> 00:44:54,550 It can deal with it can add two approximate things. 1112 00:44:54,550 --> 00:44:56,183 But you might really lose your mind 1113 00:44:56,183 --> 00:44:57,850 if you tried to do a whole string of it. 1114 00:44:57,850 --> 00:44:58,370 Yeah? 1115 00:44:58,370 --> 00:44:59,762 Yeah? 1116 00:44:59,762 --> 00:45:01,720 AUDIENCE: If he was working on the same digits, 1117 00:45:01,720 --> 00:45:05,260 like maybe seven plus seven or seven minus seven, 1118 00:45:05,260 --> 00:45:08,860 expect him to maybe do that fairly easily 1119 00:45:08,860 --> 00:45:12,010 if that's the case, right? 1120 00:45:12,010 --> 00:45:13,390 NANCY KANWISHER: Say more. 1121 00:45:13,390 --> 00:45:16,240 AUDIENCE: If it's a case that his approximate-- 1122 00:45:16,240 --> 00:45:17,510 NANCY KANWISHER: Yeah, yeah. 1123 00:45:17,510 --> 00:45:18,010 Yeah. 1124 00:45:18,010 --> 00:45:19,600 AUDIENCE: He should be able to do seven minus seven 1125 00:45:19,600 --> 00:45:20,100 fairly easy. 1126 00:45:20,100 --> 00:45:23,268 Because you know that when you subtract the same things, 1127 00:45:23,268 --> 00:45:24,310 you're going to get zero. 1128 00:45:24,310 --> 00:45:24,760 NANCY KANWISHER: Yes. 1129 00:45:24,760 --> 00:45:26,170 But it's an interesting question, actually, 1130 00:45:26,170 --> 00:45:28,000 whether that would be part of that system 1131 00:45:28,000 --> 00:45:31,700 or whether that's kind of more abstract formal thing you 1132 00:45:31,700 --> 00:45:32,200 learn. 1133 00:45:32,200 --> 00:45:34,607 So I think it depends how you do it, right? 1134 00:45:34,607 --> 00:45:35,440 So one of the ways-- 1135 00:45:35,440 --> 00:45:36,482 I didn't talk about this. 1136 00:45:36,482 --> 00:45:39,940 But those same experiments adding, say, 1137 00:45:39,940 --> 00:45:43,660 adding dots to dots, those were also done with little kids. 1138 00:45:43,660 --> 00:45:45,622 And there, what you do is you show-- 1139 00:45:45,622 --> 00:45:47,080 I don't really remember what it is. 1140 00:45:47,080 --> 00:45:48,880 But you show some array of things, 1141 00:45:48,880 --> 00:45:50,320 and you hide it behind a screen. 1142 00:45:50,320 --> 00:45:52,270 And then you show another array and hide it behind the screen. 1143 00:45:52,270 --> 00:45:53,562 And then you reveal the screen. 1144 00:45:53,562 --> 00:45:55,750 Like how many things are there? 1145 00:45:55,750 --> 00:45:59,800 That kind of stuff works spontaneously. 1146 00:45:59,800 --> 00:46:02,175 So it might tap into that system. 1147 00:46:02,175 --> 00:46:03,800 I think that's an interesting question. 1148 00:46:03,800 --> 00:46:06,250 I'm not totally sure how it would go. 1149 00:46:06,250 --> 00:46:06,750 Yeah? 1150 00:46:06,750 --> 00:46:08,125 AUDIENCE: So the second person is 1151 00:46:08,125 --> 00:46:09,450 bad at recall across the board? 1152 00:46:09,450 --> 00:46:10,710 Or is it just with numbers? 1153 00:46:10,710 --> 00:46:11,790 NANCY KANWISHER: Just with numbers. 1154 00:46:11,790 --> 00:46:12,180 Yeah. 1155 00:46:12,180 --> 00:46:13,590 I mean, there's always a little bit messy. 1156 00:46:13,590 --> 00:46:15,990 The patient literature is always like some other random stuff. 1157 00:46:15,990 --> 00:46:17,470 And how do you account for that? 1158 00:46:17,470 --> 00:46:19,110 And there's lesions in other places. 1159 00:46:19,110 --> 00:46:21,450 But to a first approximation, these 1160 00:46:21,450 --> 00:46:24,585 are reasonably number-specific deficits. 1161 00:46:24,585 --> 00:46:26,190 All right? 1162 00:46:26,190 --> 00:46:29,098 OK, so that's a bit of a hint from 1163 00:46:29,098 --> 00:46:30,390 the neuropsychology literature. 1164 00:46:30,390 --> 00:46:33,600 But there's mainly these two patients 1165 00:46:33,600 --> 00:46:36,420 and some other like messier ones. 1166 00:46:36,420 --> 00:46:40,710 And so one wants to use neuroimaging 1167 00:46:40,710 --> 00:46:42,140 to get a better picture of it. 1168 00:46:42,140 --> 00:46:44,140 Of course, that's been going on for a long time. 1169 00:46:44,140 --> 00:46:47,520 And so here's one of the early papers from Stan Dehaene's lab. 1170 00:46:47,520 --> 00:46:50,070 This is a top view of the brain. 1171 00:46:50,070 --> 00:46:51,480 So this is this parietal zone. 1172 00:46:51,480 --> 00:46:54,090 And this is what is often referred 1173 00:46:54,090 --> 00:46:58,200 to as the horizontal segment of the intraparietal sulcus. 1174 00:46:58,200 --> 00:46:59,402 hIPS to its friends. 1175 00:46:59,402 --> 00:47:01,860 And it's that sulcus I talked about that goes up like this. 1176 00:47:01,860 --> 00:47:02,970 It kind of curves over. 1177 00:47:02,970 --> 00:47:04,650 And it's like this bit right there. 1178 00:47:04,650 --> 00:47:05,490 OK? 1179 00:47:05,490 --> 00:47:07,035 That little orange strip. 1180 00:47:07,035 --> 00:47:09,660 And so what he's saying in this review article from a long time 1181 00:47:09,660 --> 00:47:12,090 ago is that that region is activated only 1182 00:47:12,090 --> 00:47:13,470 when you do calculation. 1183 00:47:13,470 --> 00:47:16,170 He means basic arithmetic in this case. 1184 00:47:16,170 --> 00:47:20,060 Not when you do all these other things. 1185 00:47:20,060 --> 00:47:23,990 But when this paper came out, I'm like, yeah, right. 1186 00:47:23,990 --> 00:47:25,850 I don't think so. 1187 00:47:25,850 --> 00:47:28,910 I can't tell you how many experiments I've run and seen 1188 00:47:28,910 --> 00:47:32,510 big ass activations right there on tasks that have nothing 1189 00:47:32,510 --> 00:47:33,470 to do with numbers. 1190 00:47:33,470 --> 00:47:34,280 So looks good. 1191 00:47:34,280 --> 00:47:35,300 Sounded good. 1192 00:47:35,300 --> 00:47:37,070 He got away with it for a while. 1193 00:47:37,070 --> 00:47:38,110 And it's not true. 1194 00:47:38,110 --> 00:47:38,873 Yeah? 1195 00:47:38,873 --> 00:47:40,790 AUDIENCE: So is the reason sort of high enough 1196 00:47:40,790 --> 00:47:41,623 that you can zap it? 1197 00:47:41,623 --> 00:47:42,770 NANCY KANWISHER: Terrible. 1198 00:47:42,770 --> 00:47:44,240 Being filmed too. 1199 00:47:44,240 --> 00:47:45,680 He's a really smart, nice guy. 1200 00:47:45,680 --> 00:47:47,972 I just like when people are a little bit fast and loose 1201 00:47:47,972 --> 00:47:50,960 and make a big claim, which you can tell at the time 1202 00:47:50,960 --> 00:47:51,785 isn't quite right. 1203 00:47:51,785 --> 00:47:52,910 It's a little bit annoying. 1204 00:47:52,910 --> 00:47:53,210 Anyway. 1205 00:47:53,210 --> 00:47:53,710 Sorry. 1206 00:47:53,710 --> 00:47:54,230 Go ahead. 1207 00:47:54,230 --> 00:47:54,500 AUDIENCE: Yeah. 1208 00:47:54,500 --> 00:47:56,577 Is the region high enough that you can zap it? 1209 00:47:56,577 --> 00:47:57,410 NANCY KANWISHER: Ah. 1210 00:47:57,410 --> 00:47:58,243 We're getting there. 1211 00:47:58,243 --> 00:48:00,410 Yes, indeed, you can. 1212 00:48:00,410 --> 00:48:02,330 But let's do a little more basic stuff first. 1213 00:48:02,330 --> 00:48:05,630 OK, so the claim is that this hIPS thing 1214 00:48:05,630 --> 00:48:07,850 is the locus of the approximate number system. 1215 00:48:07,850 --> 00:48:09,120 That was the early claim. 1216 00:48:09,120 --> 00:48:09,620 OK. 1217 00:48:12,380 --> 00:48:15,800 And for further, the claim implicit in this article 1218 00:48:15,800 --> 00:48:17,810 in this figure is that it's involved 1219 00:48:17,810 --> 00:48:20,150 in numerical representations only, 1220 00:48:20,150 --> 00:48:24,290 not any of these other things, grasping tasks, manual tasks, 1221 00:48:24,290 --> 00:48:27,230 eye-movement tasks, et cetera, et cetera, et cetera. 1222 00:48:27,230 --> 00:48:31,070 OK, really? 1223 00:48:31,070 --> 00:48:33,680 And as I mentioned, like me and lots of other people 1224 00:48:33,680 --> 00:48:36,780 had seen it looks like the same regions activated 1225 00:48:36,780 --> 00:48:38,930 in all kinds of other situations, especially 1226 00:48:38,930 --> 00:48:42,620 those involving reasoning about spatial location. 1227 00:48:42,620 --> 00:48:45,823 You guys got short shrift six weeks ago. 1228 00:48:45,823 --> 00:48:47,990 I meant to talk about the parietal lobe and its role 1229 00:48:47,990 --> 00:48:48,885 in high-level vision. 1230 00:48:48,885 --> 00:48:50,510 And it just somehow went by the boards. 1231 00:48:50,510 --> 00:48:54,110 But all this stuff is involved in aspects 1232 00:48:54,110 --> 00:48:58,040 of vision, particularly spatial vision, knowing what is where. 1233 00:48:58,040 --> 00:48:59,690 OK? 1234 00:48:59,690 --> 00:49:02,190 And so there's an alternate view, 1235 00:49:02,190 --> 00:49:06,140 which is that there's no specific brain region that's 1236 00:49:06,140 --> 00:49:10,760 specifically all only involved in discrete number per se. 1237 00:49:10,760 --> 00:49:15,410 Instead, there's a common region for processing magnitude 1238 00:49:15,410 --> 00:49:18,560 of almost any dimension, whether discrete or continuous, 1239 00:49:18,560 --> 00:49:21,470 right, that approximate number system or your exact number 1240 00:49:21,470 --> 00:49:25,570 system, and that it builds on previous representations 1241 00:49:25,570 --> 00:49:26,920 of space. 1242 00:49:26,920 --> 00:49:28,260 OK? 1243 00:49:28,260 --> 00:49:31,380 For example, the number line, right? 1244 00:49:31,380 --> 00:49:34,950 So you guys read this article for last night. 1245 00:49:34,950 --> 00:49:36,990 And just to review what the key point was, 1246 00:49:36,990 --> 00:49:40,740 this is, again, the kind of aerial view 1247 00:49:40,740 --> 00:49:42,120 with the parietal lobe here. 1248 00:49:42,120 --> 00:49:47,820 And that's the hIPS region, yeah, 1249 00:49:47,820 --> 00:49:49,410 that was in the previous slide. 1250 00:49:49,410 --> 00:49:51,870 And you can see it's this horizontal part of that sulcus 1251 00:49:51,870 --> 00:49:53,430 way up in the parietal lobe. 1252 00:49:53,430 --> 00:49:56,100 And the yellow and green means that there's 1253 00:49:56,100 --> 00:49:59,370 overlapping activation for both symbolic calculation, that's 1254 00:49:59,370 --> 00:50:02,430 like with symbols, and for nonsymbolic calculation. 1255 00:50:02,430 --> 00:50:04,870 That's like dot arrays stuff like that, right? 1256 00:50:04,870 --> 00:50:08,370 And so it's activated for both of those. 1257 00:50:08,370 --> 00:50:13,950 And the point of this paper is, first of all, 1258 00:50:13,950 --> 00:50:18,900 that there's also overlap with the eye-movement system, right? 1259 00:50:18,900 --> 00:50:21,000 And so here, they're really asking, 1260 00:50:21,000 --> 00:50:23,340 is this spatial representation kind 1261 00:50:23,340 --> 00:50:26,310 of co-opted in your representation of number 1262 00:50:26,310 --> 00:50:28,350 using a kind of spatial number line, right? 1263 00:50:28,350 --> 00:50:29,700 It makes perfect sense. 1264 00:50:29,700 --> 00:50:31,650 Animals need a representation of space. 1265 00:50:31,650 --> 00:50:33,780 It's like extremely basic, right? 1266 00:50:33,780 --> 00:50:37,080 And once you have that, you can co-opt it and represent numbers 1267 00:50:37,080 --> 00:50:39,660 in that same spatial code. 1268 00:50:39,660 --> 00:50:43,330 And as you guys all read, the cool result from that paper, 1269 00:50:43,330 --> 00:50:47,250 which is also from Stan Dehaene's lab, 1270 00:50:47,250 --> 00:50:51,570 is that when you take that region right in there, 1271 00:50:51,570 --> 00:50:54,390 you take those voxels in there, and you train them 1272 00:50:54,390 --> 00:50:57,840 on making leftward versus rightward saccades. 1273 00:50:57,840 --> 00:50:59,700 So now you have a classifier that 1274 00:50:59,700 --> 00:51:01,560 looks at the pattern of activation there, 1275 00:51:01,560 --> 00:51:02,940 can distinguish a leftward versus 1276 00:51:02,940 --> 00:51:04,050 a rightward versus saccade. 1277 00:51:04,050 --> 00:51:05,050 I'm just reviewing this. 1278 00:51:05,050 --> 00:51:07,110 Hopefully it was clear enough. 1279 00:51:07,110 --> 00:51:10,200 That same classifier can then distinguish 1280 00:51:10,200 --> 00:51:12,510 subtraction versus addition. 1281 00:51:12,510 --> 00:51:14,430 Did you guys all get that from the paper? 1282 00:51:14,430 --> 00:51:14,930 Yeah? 1283 00:51:14,930 --> 00:51:17,170 It's pretty cool, isn't it? 1284 00:51:17,170 --> 00:51:18,990 Anyway, so that's kind of nice evidence 1285 00:51:18,990 --> 00:51:20,790 that the same spatial system that's 1286 00:51:20,790 --> 00:51:23,070 used in spatial attention and eye movements 1287 00:51:23,070 --> 00:51:28,850 has been co-opted to represent numbers as well. 1288 00:51:28,850 --> 00:51:31,700 OK. 1289 00:51:31,700 --> 00:51:35,300 All right, so I just wanted to incorporate that. 1290 00:51:35,300 --> 00:51:37,520 In case anybody missed what the paper was about, 1291 00:51:37,520 --> 00:51:39,800 those were the key points. 1292 00:51:39,800 --> 00:51:42,710 Other early studies have asked more directly 1293 00:51:42,710 --> 00:51:45,890 this question of whether different kinds of magnitude 1294 00:51:45,890 --> 00:51:48,170 are all represented together in the brain. 1295 00:51:48,170 --> 00:51:51,350 And this study is quite clever. 1296 00:51:51,350 --> 00:51:55,280 They used a variant of the fact that I showed you guys before. 1297 00:51:55,280 --> 00:51:57,230 Remember when saying whether the number is 1298 00:51:57,230 --> 00:51:58,940 greater or less than 65, it's harder 1299 00:51:58,940 --> 00:52:01,970 when it's closer to 65 than when it's farther from 65. 1300 00:52:01,970 --> 00:52:04,310 OK, even though I was showing you symbols, 1301 00:52:04,310 --> 00:52:05,780 that was key thing, right? 1302 00:52:05,780 --> 00:52:08,570 So that's called the distance effect, right? 1303 00:52:08,570 --> 00:52:10,160 And that's true for all comparisons. 1304 00:52:10,160 --> 00:52:12,980 And so this study exploits that distance effect. 1305 00:52:12,980 --> 00:52:15,560 And they use stimuli like this. 1306 00:52:15,560 --> 00:52:19,020 And they ask, which one is larger? 1307 00:52:19,020 --> 00:52:21,440 And it could be larger in absolute size. 1308 00:52:21,440 --> 00:52:23,360 Like the two is larger here. 1309 00:52:23,360 --> 00:52:25,370 Or it can be larger in number meaning 1310 00:52:25,370 --> 00:52:26,990 like the seven is larger. 1311 00:52:26,990 --> 00:52:29,030 OK, so in different blocks, you're saying, 1312 00:52:29,030 --> 00:52:30,590 which one is physically larger? 1313 00:52:30,590 --> 00:52:32,990 Which one is numerically larger? 1314 00:52:32,990 --> 00:52:34,490 Which one is brighter? 1315 00:52:34,490 --> 00:52:36,470 That would be this one here. 1316 00:52:36,470 --> 00:52:41,120 And then they just have a control with letters. 1317 00:52:41,120 --> 00:52:42,170 OK? 1318 00:52:42,170 --> 00:52:43,580 And so then-- sorry. 1319 00:52:43,580 --> 00:52:45,110 The design is slightly complicated. 1320 00:52:45,110 --> 00:52:48,080 So there's these three main tasks and a control task. 1321 00:52:48,080 --> 00:52:50,900 But then within each, they have the difficult version 1322 00:52:50,900 --> 00:52:52,300 and the easy version. 1323 00:52:52,300 --> 00:52:54,800 And the difficult version is when the comparisons are close, 1324 00:52:54,800 --> 00:52:57,140 two similar brightnesses, two similar numbers, 1325 00:52:57,140 --> 00:53:00,030 two similar sizes versus two larger ones. 1326 00:53:00,030 --> 00:53:00,530 OK? 1327 00:53:00,530 --> 00:53:02,990 So that's what all this garbage shows. 1328 00:53:02,990 --> 00:53:05,047 OK, so then you do that subtraction. 1329 00:53:05,047 --> 00:53:07,130 You look, and you say, OK, what parts of the brain 1330 00:53:07,130 --> 00:53:11,000 are more active when you do the difficult versus easy number 1331 00:53:11,000 --> 00:53:12,540 comparison? 1332 00:53:12,540 --> 00:53:14,730 Like saying, which is larger? 1333 00:53:14,730 --> 00:53:17,640 It's not that difficult. But two versus three 1334 00:53:17,640 --> 00:53:19,860 versus two versus seven. 1335 00:53:19,860 --> 00:53:21,780 OK? 1336 00:53:21,780 --> 00:53:26,160 And so what they find is that similar regions of the brain 1337 00:53:26,160 --> 00:53:29,680 are active for all three of those kinds of comparisons. 1338 00:53:29,680 --> 00:53:30,180 OK? 1339 00:53:30,180 --> 00:53:34,530 So it's not like you get just one for symbolic number 1340 00:53:34,530 --> 00:53:38,370 or for the two magnitude tasks. 1341 00:53:38,370 --> 00:53:40,140 All of those different kinds of magnitude 1342 00:53:40,140 --> 00:53:43,440 activate the same regions. 1343 00:53:43,440 --> 00:53:51,750 And so the conclusion is that number and size and brightness 1344 00:53:51,750 --> 00:53:55,890 engage a common parietal spatial code, OK, an overlapping 1345 00:53:55,890 --> 00:53:57,390 region for all of these. 1346 00:53:57,390 --> 00:53:59,860 Does that make sense? 1347 00:53:59,860 --> 00:54:01,830 OK. 1348 00:54:01,830 --> 00:54:05,670 And so that shows, in this case, that it's not just 1349 00:54:05,670 --> 00:54:10,030 symbolic number but also magnitude. 1350 00:54:10,030 --> 00:54:11,910 Which one is bigger, right? 1351 00:54:11,910 --> 00:54:14,910 It's kind of continuous magnitude idea. 1352 00:54:14,910 --> 00:54:17,130 OK? 1353 00:54:17,130 --> 00:54:19,480 OK. 1354 00:54:19,480 --> 00:54:19,980 Right. 1355 00:54:19,980 --> 00:54:23,190 So one worry is that, in each of these cases, 1356 00:54:23,190 --> 00:54:27,590 they're comparing a difficult condition to an easy condition. 1357 00:54:27,590 --> 00:54:30,020 And so maybe the regions they got 1358 00:54:30,020 --> 00:54:34,030 are just engaged in any kind of task difficulty. 1359 00:54:34,030 --> 00:54:39,070 Maybe if they had done a syntactic task 1360 00:54:39,070 --> 00:54:41,978 on language stimuli that was difficult versus easy, 1361 00:54:41,978 --> 00:54:43,270 they would get the same things. 1362 00:54:43,270 --> 00:54:44,950 From this experiment, we don't know. 1363 00:54:44,950 --> 00:54:47,080 We'll talk more about that in a couple weeks when 1364 00:54:47,080 --> 00:54:50,650 we talk about language, right? 1365 00:54:50,650 --> 00:54:54,070 But here's at least one control that deals with that sort of 1366 00:54:54,070 --> 00:54:57,850 and which does a TMS experiment, as you suggested a while back. 1367 00:54:57,850 --> 00:54:59,530 OK, so this is kind of cool experiment. 1368 00:54:59,530 --> 00:55:01,780 I mean, it's weird, but sort of cool. 1369 00:55:01,780 --> 00:55:02,830 OK, so what do they do? 1370 00:55:02,830 --> 00:55:06,190 They use-- OK, so they have, again, 1371 00:55:06,190 --> 00:55:09,100 an easy task and a hard task. 1372 00:55:09,100 --> 00:55:12,700 Again, it's the thing greater or less than 65. 1373 00:55:12,700 --> 00:55:14,593 Not very hard, right? 1374 00:55:14,593 --> 00:55:15,760 The hard one it's that hard. 1375 00:55:15,760 --> 00:55:19,030 But so is it greater or less than 65? 1376 00:55:19,030 --> 00:55:23,230 And it's either a symbolic number, or it's a dot array. 1377 00:55:23,230 --> 00:55:24,730 You can't really see it, but there's 1378 00:55:24,730 --> 00:55:27,220 a bunch of teeny dots in there. 1379 00:55:27,220 --> 00:55:29,020 Or in the other condition, they have 1380 00:55:29,020 --> 00:55:34,270 to say whether that ellipse is more horizontal or vertical. 1381 00:55:34,270 --> 00:55:35,347 OK? 1382 00:55:35,347 --> 00:55:37,930 And so you spend a lot of time, before you run the experiment, 1383 00:55:37,930 --> 00:55:42,610 measuring reaction time and accuracy to balance difficulty 1384 00:55:42,610 --> 00:55:44,920 within the easy conditions and balance difficulty 1385 00:55:44,920 --> 00:55:46,660 within the hard conditions. 1386 00:55:46,660 --> 00:55:48,350 OK? 1387 00:55:48,350 --> 00:55:53,260 So then what they do is they do something called offline TMS. 1388 00:55:53,260 --> 00:55:54,040 OK? 1389 00:55:54,040 --> 00:55:57,820 Offline TMS, I didn't talk about this much before. 1390 00:55:57,820 --> 00:56:00,923 The standard kinds of TMS, you stick the coil right 1391 00:56:00,923 --> 00:56:01,840 on the subject's head. 1392 00:56:01,840 --> 00:56:03,910 There's a subject doing a task on a monitor. 1393 00:56:03,910 --> 00:56:05,910 And somebody is standing there holding the coil. 1394 00:56:05,910 --> 00:56:08,170 It's really kind of rudimentary. 1395 00:56:08,170 --> 00:56:11,230 And right at a key point of the trial, 1396 00:56:11,230 --> 00:56:13,850 you deliver a zap to disrupt that part of the brain. 1397 00:56:13,850 --> 00:56:16,160 And you find out how much that interferes 1398 00:56:16,160 --> 00:56:17,410 with performance on that task. 1399 00:56:17,410 --> 00:56:20,620 That's the standard online kind of TMS thing. 1400 00:56:20,620 --> 00:56:23,260 But there's also offline TMS, where 1401 00:56:23,260 --> 00:56:28,420 you zap people at a slow rate for like 10 minutes. 1402 00:56:28,420 --> 00:56:31,150 And then the idea is that you've kind of generally 1403 00:56:31,150 --> 00:56:35,740 disrupted that piece of brain for, say, another 10 minutes. 1404 00:56:35,740 --> 00:56:37,030 It's a little bit scarier. 1405 00:56:37,030 --> 00:56:40,060 But it's just like 10 minutes, right? 1406 00:56:40,060 --> 00:56:41,620 OK, and so that way, you don't have 1407 00:56:41,620 --> 00:56:43,810 to be quite so fancy about the precise timing. 1408 00:56:43,810 --> 00:56:46,480 You can just kind of reduce its effectiveness 1409 00:56:46,480 --> 00:56:47,590 for a whole 10 minutes. 1410 00:56:47,590 --> 00:56:50,650 OK, so that's what they did here, offline TMS. 1411 00:56:50,650 --> 00:56:53,830 So you sit there and get zapped for 10 minutes slowly here. 1412 00:56:53,830 --> 00:56:56,260 And then you do some math tasks. 1413 00:56:56,260 --> 00:56:57,280 OK. 1414 00:56:57,280 --> 00:57:01,000 OK, so what they find is that zapping 1415 00:57:01,000 --> 00:57:06,070 the left intraparietal sulcus disrupts the magnitude tasks 1416 00:57:06,070 --> 00:57:08,560 on both numbers and dots. 1417 00:57:08,560 --> 00:57:11,410 But it doesn't mess up the shape tasks with the ellipses, 1418 00:57:11,410 --> 00:57:14,720 even though the ellipses are balanced for difficulty. 1419 00:57:14,720 --> 00:57:15,680 OK? 1420 00:57:15,680 --> 00:57:17,630 So that's at least a little bit of an argument 1421 00:57:17,630 --> 00:57:20,180 that it's not just about generic difficulty, 1422 00:57:20,180 --> 00:57:22,230 at least in this experiment. 1423 00:57:22,230 --> 00:57:24,240 OK? 1424 00:57:24,240 --> 00:57:27,340 All right, I think that's what I just said. 1425 00:57:27,340 --> 00:57:30,220 So that's some evidence for a role 1426 00:57:30,220 --> 00:57:32,380 of at least the left intraparietal sulcus 1427 00:57:32,380 --> 00:57:34,692 in both symbolic and nonsymbolic number. 1428 00:57:34,692 --> 00:57:36,400 Again, nonsymbolic number just means dots 1429 00:57:36,400 --> 00:57:40,120 without Arabic numbers, not just any difficulty. 1430 00:57:40,120 --> 00:57:42,280 All right, so that's all very nice. 1431 00:57:42,280 --> 00:57:44,050 But it's crude as hell, right? 1432 00:57:44,050 --> 00:57:46,030 We found these big, blurry chunks of brain 1433 00:57:46,030 --> 00:57:46,960 that are implicated. 1434 00:57:46,960 --> 00:57:49,060 And we zapped a big chunk of brain 1435 00:57:49,060 --> 00:57:50,980 and slightly reduced performance. 1436 00:57:50,980 --> 00:57:52,450 It's like, OK, better than nothing. 1437 00:57:52,450 --> 00:57:54,100 But it's not very impressive. 1438 00:57:54,100 --> 00:57:57,260 What are the actual neurons doing in the brain? 1439 00:57:57,260 --> 00:57:59,800 Well, now it becomes really important and useful 1440 00:57:59,800 --> 00:58:02,710 that this approximate number system 1441 00:58:02,710 --> 00:58:06,160 that I've been talking about is also present in animals. 1442 00:58:06,160 --> 00:58:08,410 And that means we can use animal models. 1443 00:58:08,410 --> 00:58:10,990 And we can record from individual neurons 1444 00:58:10,990 --> 00:58:13,000 in the parietal lobes of monkeys when 1445 00:58:13,000 --> 00:58:16,630 they do number tasks to find out what actual neurons are doing. 1446 00:58:16,630 --> 00:58:17,500 OK? 1447 00:58:17,500 --> 00:58:19,960 And so there's a guy named Andreas Nieder, who's been 1448 00:58:19,960 --> 00:58:22,370 doing this for a long time. 1449 00:58:22,370 --> 00:58:24,800 And he has some pretty remarkable data. 1450 00:58:24,800 --> 00:58:29,900 And so he starts by training monkeys to do a number task. 1451 00:58:29,900 --> 00:58:31,510 So here's what the monkey sees. 1452 00:58:31,510 --> 00:58:35,320 Monkey sees a sample, some number of dots. 1453 00:58:35,320 --> 00:58:38,440 And then there's a memory delay, in this case, one second. 1454 00:58:38,440 --> 00:58:40,000 And then he has to do a matching task 1455 00:58:40,000 --> 00:58:42,770 and choose that array, not this array. 1456 00:58:42,770 --> 00:58:43,270 OK? 1457 00:58:43,270 --> 00:58:45,228 So he's got to remember that there's three dots 1458 00:58:45,228 --> 00:58:46,570 and choose the right three. 1459 00:58:46,570 --> 00:58:49,090 And notice that the sizes and configuration of the dots 1460 00:58:49,090 --> 00:58:49,640 have changed. 1461 00:58:49,640 --> 00:58:51,940 So we have to do something more like remember 1462 00:58:51,940 --> 00:58:56,950 three in whatever mental monkey E's version of three exists. 1463 00:58:56,950 --> 00:58:58,060 OK? 1464 00:58:58,060 --> 00:59:00,670 OK, simple matching tasks. 1465 00:59:00,670 --> 00:59:03,580 Then he records from neurons in the parietal and frontal cortex 1466 00:59:03,580 --> 00:59:05,660 in monkeys. 1467 00:59:05,660 --> 00:59:10,040 And he finds neurons that are sort of specific for number. 1468 00:59:10,040 --> 00:59:13,380 OK, so here's time in that task. 1469 00:59:13,380 --> 00:59:17,570 This is the time that the sample is presented right here. 1470 00:59:17,570 --> 00:59:21,680 And here is the response of a single neuron that likes 1471 00:59:21,680 --> 00:59:26,400 two more than anything else. 1472 00:59:26,400 --> 00:59:27,390 OK? 1473 00:59:27,390 --> 00:59:30,540 And that too, notice, is all different kinds 1474 00:59:30,540 --> 00:59:33,780 of spatial arrangements and sizes of the dots. 1475 00:59:33,780 --> 00:59:37,770 What's common about all of them is that it's two. 1476 00:59:37,770 --> 00:59:40,570 Next best, it likes four. 1477 00:59:40,570 --> 00:59:41,070 OK? 1478 00:59:41,070 --> 00:59:43,180 And it generalizes across number from there. 1479 00:59:43,180 --> 00:59:44,520 So it's approximate. 1480 00:59:44,520 --> 00:59:47,640 It's not like high for two and zero for everything else. 1481 00:59:47,640 --> 00:59:50,550 It's got a kind of generalization gradient. 1482 00:59:50,550 --> 00:59:52,060 But it prefers two. 1483 00:59:52,060 --> 00:59:52,560 OK? 1484 00:59:52,560 --> 00:59:54,630 So that's a number neuron. 1485 00:59:54,630 --> 00:59:56,100 Yeah? 1486 00:59:56,100 --> 01:00:01,270 OK, here's a six neuron. 1487 01:00:01,270 --> 01:00:03,790 This neuron likes six. 1488 01:00:03,790 --> 01:00:07,840 Here it is same task during presentation of trials here. 1489 01:00:07,840 --> 01:00:09,766 Red is six. 1490 01:00:09,766 --> 01:00:13,390 Next closest is like eight and maybe 10. 1491 01:00:13,390 --> 01:00:16,360 So it also generalizes as well, but it responds more to six 1492 01:00:16,360 --> 01:00:17,650 than anything else. 1493 01:00:17,650 --> 01:00:18,550 Pretty awesome. 1494 01:00:18,550 --> 01:00:19,940 Huh? 1495 01:00:19,940 --> 01:00:25,370 OK, now that doesn't tell us how it was computed, right? 1496 01:00:25,370 --> 01:00:28,970 So finding a single neuron that does something spectacular 1497 01:00:28,970 --> 01:00:29,660 is thrilling. 1498 01:00:29,660 --> 01:00:30,380 We all love it. 1499 01:00:30,380 --> 01:00:32,270 It's great fun. 1500 01:00:32,270 --> 01:00:34,400 And we're closer to the neural circuit 1501 01:00:34,400 --> 01:00:37,160 because we found a neuron that seems to be part of the action. 1502 01:00:37,160 --> 01:00:40,340 But notice it doesn't tell us how that neuron made 1503 01:00:40,340 --> 01:00:42,620 that computation, right? 1504 01:00:42,620 --> 01:00:45,290 What are the circuits that led into it, that enabled it 1505 01:00:45,290 --> 01:00:48,285 to be specific to six or two? 1506 01:00:48,285 --> 01:00:50,450 But it's still cool. 1507 01:00:50,450 --> 01:00:53,850 OK, but next, we want to know, how abstract are those neurons? 1508 01:00:53,850 --> 01:00:56,210 This is just dot arrays. 1509 01:00:56,210 --> 01:00:57,840 OK? 1510 01:00:57,840 --> 01:01:01,110 And they're just presented in one array. 1511 01:01:01,110 --> 01:01:04,740 So next, Andreas Nieder trains his monkeys 1512 01:01:04,740 --> 01:01:08,670 to keep track of the number of things that happen over time. 1513 01:01:08,670 --> 01:01:10,290 It's not a spatial array. 1514 01:01:10,290 --> 01:01:12,600 It's a temporal sequence. 1515 01:01:12,600 --> 01:01:13,230 OK? 1516 01:01:13,230 --> 01:01:16,870 So we have to see that there's four things coming in here 1517 01:01:16,870 --> 01:01:20,460 and then choose the array that matches with four. 1518 01:01:20,460 --> 01:01:22,800 OK? 1519 01:01:22,800 --> 01:01:25,660 See how this is the way to ask how abstract those number 1520 01:01:25,660 --> 01:01:26,280 neurons are. 1521 01:01:26,280 --> 01:01:29,505 Are they really representing the abstract magnitude of two 1522 01:01:29,505 --> 01:01:30,700 or six or whatever it is. 1523 01:01:30,700 --> 01:01:32,117 Or are they representing something 1524 01:01:32,117 --> 01:01:35,410 about the shape of a two-type array or a six-type array. 1525 01:01:35,410 --> 01:01:40,540 OK, and they can also test over different modalities. 1526 01:01:40,540 --> 01:01:43,530 So now they present four different tones. 1527 01:01:43,530 --> 01:01:46,780 And the monkey has to choose the four dots. 1528 01:01:46,780 --> 01:01:47,280 OK? 1529 01:01:47,280 --> 01:01:51,690 Now it's both over time and over sensory modality. 1530 01:01:51,690 --> 01:01:54,210 So how abstract are those number neurons? 1531 01:01:54,210 --> 01:01:56,650 OK, they're pretty abstract. 1532 01:01:56,650 --> 01:01:58,270 So here are a few number neurons. 1533 01:01:58,270 --> 01:02:00,300 Cell one is in the blue colors. 1534 01:02:00,300 --> 01:02:03,750 And here is its response in light blue to dots, 1535 01:02:03,750 --> 01:02:06,660 one dot, two dots, three dots, four dots. 1536 01:02:06,660 --> 01:02:12,120 And here is the same cell responding to sounds. 1537 01:02:12,120 --> 01:02:16,605 It's specific to one, both for sounds and dot arrays. 1538 01:02:16,605 --> 01:02:18,000 Isn't that cool? 1539 01:02:18,000 --> 01:02:22,410 And you see the green cell is selected for two, 1540 01:02:22,410 --> 01:02:26,440 whether in dots or sounds, and so forth. 1541 01:02:26,440 --> 01:02:27,940 Pretty cool, huh? 1542 01:02:27,940 --> 01:02:31,706 So these are very abstract number neurons. 1543 01:02:31,706 --> 01:02:33,350 Does that makes sense? 1544 01:02:33,350 --> 01:02:35,420 OK. 1545 01:02:35,420 --> 01:02:38,110 OK. 1546 01:02:38,110 --> 01:02:41,590 OK, now these monkeys are trained on number tasks. 1547 01:02:41,590 --> 01:02:44,230 So you might think that these kinds of abstract number 1548 01:02:44,230 --> 01:02:46,480 neurons-- and they're trained to do the generalization 1549 01:02:46,480 --> 01:02:47,950 from tones to arrays. 1550 01:02:47,950 --> 01:02:50,950 So maybe those neurons wouldn't live in their brains 1551 01:02:50,950 --> 01:02:53,712 if they hadn't been trained to do that. 1552 01:02:53,712 --> 01:02:55,670 But I don't have time to show you all the data. 1553 01:02:55,670 --> 01:02:59,650 But in subsequent work, the same team 1554 01:02:59,650 --> 01:03:02,260 has recorded from monkeys before any training. 1555 01:03:02,260 --> 01:03:06,200 And you find similar number of neurons. 1556 01:03:06,200 --> 01:03:10,570 So it does seem like these are things that exist in-- 1557 01:03:10,570 --> 01:03:13,010 and remember that's consistent with what I said before, 1558 01:03:13,010 --> 01:03:17,710 which is that a lot of these number abilities 1559 01:03:17,710 --> 01:03:20,830 are present in animals without any training and in newborns. 1560 01:03:20,830 --> 01:03:22,900 And so it makes sense that some of those neurons 1561 01:03:22,900 --> 01:03:25,300 would be around even in advance of any training. 1562 01:03:25,300 --> 01:03:27,550 AUDIENCE: How many neurons did they have to look at it 1563 01:03:27,550 --> 01:03:28,330 to find? 1564 01:03:28,330 --> 01:03:29,410 NANCY KANWISHER: Oh, that's a good question. 1565 01:03:29,410 --> 01:03:30,760 I forget what percent it is. 1566 01:03:30,760 --> 01:03:32,830 We could look it up in the Nieder paper. 1567 01:03:32,830 --> 01:03:33,700 Yeah. 1568 01:03:33,700 --> 01:03:35,830 It's not like you record from thousands, 1569 01:03:35,830 --> 01:03:38,890 and you find 10, right? 1570 01:03:38,890 --> 01:03:42,910 Remember they know where to look from, first, the human lesion 1571 01:03:42,910 --> 01:03:45,940 literature and then the human functional imaging literature. 1572 01:03:45,940 --> 01:03:48,940 And then there's also monkey neuroimaging literature 1573 01:03:48,940 --> 01:03:50,860 where you can have monkeys doing dot tasks. 1574 01:03:50,860 --> 01:03:52,647 So you can know where to look. 1575 01:03:52,647 --> 01:03:53,980 Because the brain's a big place. 1576 01:03:53,980 --> 01:03:55,813 If you're just sticking electrodes all over, 1577 01:03:55,813 --> 01:03:56,650 God help you, right? 1578 01:03:56,650 --> 01:03:59,597 So they know to go up in that parietal lobe 1579 01:03:59,597 --> 01:04:01,930 if that region is homologous between humans and monkeys. 1580 01:04:01,930 --> 01:04:03,520 And there's a lot of other evidence 1581 01:04:03,520 --> 01:04:06,130 that that region is homologous. 1582 01:04:06,130 --> 01:04:07,930 So they know how to get in the right zone. 1583 01:04:07,930 --> 01:04:09,472 And I'm sure, once in the right zone, 1584 01:04:09,472 --> 01:04:10,840 they're not all number neurons. 1585 01:04:10,840 --> 01:04:12,730 I'm sure it's a relatively small percent. 1586 01:04:12,730 --> 01:04:15,384 But it's not a trivial percent. 1587 01:04:15,384 --> 01:04:16,638 Yeah? 1588 01:04:16,638 --> 01:04:19,180 AUDIENCE: Do we have sense for how fractions are represented? 1589 01:04:19,180 --> 01:04:21,105 Because all of these seem to be discrete. 1590 01:04:23,710 --> 01:04:26,003 Or [INAUDIBLE], any ideas? 1591 01:04:26,003 --> 01:04:26,920 NANCY KANWISHER: Yeah. 1592 01:04:26,920 --> 01:04:30,220 Well, it's tricky because, certainly, at the single unit 1593 01:04:30,220 --> 01:04:33,490 level, you'd have to either find some natural version where 1594 01:04:33,490 --> 01:04:36,640 monkeys think about fractions naturally 1595 01:04:36,640 --> 01:04:39,310 or teach them about fractions, which would be really hard. 1596 01:04:39,310 --> 01:04:41,740 Because, for some reason, fractions are just really hard. 1597 01:04:41,740 --> 01:04:44,800 Like all the people who study math education, 1598 01:04:44,800 --> 01:04:46,862 it's like the key problem is, how do you get 1599 01:04:46,862 --> 01:04:48,070 kids to understand fractions? 1600 01:04:48,070 --> 01:04:50,950 I don't know why they're such a tough thing. 1601 01:04:50,950 --> 01:04:53,710 But apparently, it's like a real dividing line, 1602 01:04:53,710 --> 01:04:56,980 the kids who get fractions and the kids who don't. 1603 01:04:56,980 --> 01:04:59,310 So I'd have to think. 1604 01:04:59,310 --> 01:05:04,500 But, occasionally, there are patients with electrodes 1605 01:05:04,500 --> 01:05:05,340 in their brains. 1606 01:05:05,340 --> 01:05:07,230 And one could look at that. 1607 01:05:07,230 --> 01:05:09,750 Actually, I took this slide out, but there's 1608 01:05:09,750 --> 01:05:12,570 a paper that came out last year where they found number 1609 01:05:12,570 --> 01:05:14,335 neurons in humans as well. 1610 01:05:14,335 --> 01:05:15,960 I took it out because I didn't know how 1611 01:05:15,960 --> 01:05:16,920 to integrate it in the lecture. 1612 01:05:16,920 --> 01:05:18,480 Because the number neurons are deep 1613 01:05:18,480 --> 01:05:20,910 in the medial temporal lobe, far from the parietal lobe. 1614 01:05:20,910 --> 01:05:22,800 And it's like, I don't know how that fits. 1615 01:05:22,800 --> 01:05:24,790 I don't know if that's the same thing or something else. 1616 01:05:24,790 --> 01:05:26,340 But anyway, there are at least some number neurons 1617 01:05:26,340 --> 01:05:27,820 that have been found in humans. 1618 01:05:27,820 --> 01:05:31,050 And you could, in principle, look for number neurons 1619 01:05:31,050 --> 01:05:33,420 up in the parietal lobe. 1620 01:05:33,420 --> 01:05:38,125 In fact, I have a guy I'm trying to collaborate with. 1621 01:05:38,125 --> 01:05:39,750 I'm begging him to collaborate with me. 1622 01:05:39,750 --> 01:05:42,870 He's got two people who have arrays of electrodes 1623 01:05:42,870 --> 01:05:46,410 chronically implanted right up in this region 1624 01:05:46,410 --> 01:05:49,500 here because they are paralyzed. 1625 01:05:49,500 --> 01:05:51,000 They had spinal damage. 1626 01:05:51,000 --> 01:05:52,920 And like Michael Cohen's lecture, 1627 01:05:52,920 --> 01:05:55,170 he's got arrays of electrodes where 1628 01:05:55,170 --> 01:05:57,300 he's trying to use the neural responses there 1629 01:05:57,300 --> 01:05:59,040 to direct robot arms. 1630 01:05:59,040 --> 01:06:00,570 And so there's two of these people 1631 01:06:00,570 --> 01:06:02,180 who have these chronically implanted things. 1632 01:06:02,180 --> 01:06:03,722 I'm like, oh, please, please, please, 1633 01:06:03,722 --> 01:06:05,760 can I collaborate with you and get responses 1634 01:06:05,760 --> 01:06:07,680 from your patients' neurons? 1635 01:06:07,680 --> 01:06:09,740 Was there a question over here a moment ago? 1636 01:06:09,740 --> 01:06:10,550 Sorry. 1637 01:06:10,550 --> 01:06:12,800 I thought I saw a hand go up. 1638 01:06:12,800 --> 01:06:14,990 OK, so let me wrap up. 1639 01:06:14,990 --> 01:06:17,690 So I've been arguing that this approximate number 1640 01:06:17,690 --> 01:06:21,740 system is shared with animals and newborns. 1641 01:06:21,740 --> 01:06:25,460 It's a pretty basic system that lots of animals have. 1642 01:06:25,460 --> 01:06:28,100 It follows Weber's law, which you should remember. 1643 01:06:28,100 --> 01:06:30,230 I don't like testing you guys on esoteric facts. 1644 01:06:30,230 --> 01:06:32,780 But Weber's law is a very fundamental fact. 1645 01:06:32,780 --> 01:06:36,110 And you should know it about perception and, in particular, 1646 01:06:36,110 --> 01:06:36,930 about number. 1647 01:06:36,930 --> 01:06:39,350 It tells you that the ability to discriminate two numbers 1648 01:06:39,350 --> 01:06:44,360 goes as the ratio, not as the difference of those numbers. 1649 01:06:44,360 --> 01:06:48,290 And that these approximate magnitude representations 1650 01:06:48,290 --> 01:06:52,040 measured both behaviorally and neurally in humans and animals 1651 01:06:52,040 --> 01:06:54,740 are very abstract to the particular objects, 1652 01:06:54,740 --> 01:06:56,330 to the modality, to whether they come 1653 01:06:56,330 --> 01:06:59,120 in over space or time, et cetera, 1654 01:06:59,120 --> 01:07:03,140 whether they're represented in symbols or arrays of items. 1655 01:07:03,140 --> 01:07:04,502 OK? 1656 01:07:04,502 --> 01:07:06,710 I mentioned that there are big individual differences 1657 01:07:06,710 --> 01:07:11,210 in humans in the precision of the approximate number system. 1658 01:07:11,210 --> 01:07:15,080 And that is predictive of later arithmetic abilities 1659 01:07:15,080 --> 01:07:18,170 independent of IQ. 1660 01:07:18,170 --> 01:07:21,230 And we talked about the horizontal segment 1661 01:07:21,230 --> 01:07:24,110 of the intraparietal sulcus as a key locus 1662 01:07:24,110 --> 01:07:29,120 for the approximate number system in humans, 1663 01:07:29,120 --> 01:07:32,330 including number-specific neurons. 1664 01:07:32,330 --> 01:07:34,370 And we also talked about, both in some 1665 01:07:34,370 --> 01:07:37,430 of the papers I mentioned and the paper you guys read, 1666 01:07:37,430 --> 01:07:40,910 that there seems to be that that approximate number system up 1667 01:07:40,910 --> 01:07:44,090 here in the parietal lobe, so far, doesn't seem 1668 01:07:44,090 --> 01:07:48,710 to be one of these extremely specialized systems like faces 1669 01:07:48,710 --> 01:07:51,680 and motion and navigation, which may turn out 1670 01:07:51,680 --> 01:07:53,880 to be less specialized later with pending more data. 1671 01:07:53,880 --> 01:07:55,670 But at the moment, we can already 1672 01:07:55,670 --> 01:07:58,910 see that these number representations overlap a lot 1673 01:07:58,910 --> 01:08:02,510 with representations of space, shown perhaps most dramatically 1674 01:08:02,510 --> 01:08:03,980 in the paper you guys read showing 1675 01:08:03,980 --> 01:08:07,730 cross decoding between eye-movement direction 1676 01:08:07,730 --> 01:08:10,310 and arithmetic operations. 1677 01:08:10,310 --> 01:08:12,845 OK, hang on. 1678 01:08:12,845 --> 01:08:13,970 I'm almost done summing up. 1679 01:08:13,970 --> 01:08:16,550 Number neurons, we talked about that. 1680 01:08:16,550 --> 01:08:18,859 Yes, so we'll give the last word to Stan Dehaene, who 1681 01:08:18,859 --> 01:08:21,380 started off with this very extreme view 1682 01:08:21,380 --> 01:08:25,340 and has evolved to a still interesting but slightly less 1683 01:08:25,340 --> 01:08:26,510 extreme view. 1684 01:08:26,510 --> 01:08:29,540 He says, the brain treats number like a specific category 1685 01:08:29,540 --> 01:08:33,439 of knowledge requiring its own neurological apparatus 1686 01:08:33,439 --> 01:08:35,390 in the parietal lobe. 1687 01:08:35,390 --> 01:08:38,149 But when it comes to subtler distinctions, such as number 1688 01:08:38,149 --> 01:08:42,740 versus length, space, or time, the specificity of hIPS 1689 01:08:42,740 --> 01:08:43,819 vanishes. 1690 01:08:43,819 --> 01:08:46,520 No part of hIPS appears to be involved 1691 01:08:46,520 --> 01:08:50,160 in numerical computations alone. 1692 01:08:50,160 --> 01:08:53,689 In fact, he goes further to say that the human brain, 1693 01:08:53,689 --> 01:08:57,229 in general, is neither anisotropic white paper, 1694 01:08:57,229 --> 01:09:00,859 like equipotential, where all the regions are equivalent, 1695 01:09:00,859 --> 01:09:02,779 nor a neat arrangement of tightly 1696 01:09:02,779 --> 01:09:05,750 specialized and well-separated modules. 1697 01:09:05,750 --> 01:09:06,265 All right? 1698 01:09:06,265 --> 01:09:07,640 Anyway, OK, there was a question. 1699 01:09:07,640 --> 01:09:08,555 Sorry. 1700 01:09:08,555 --> 01:09:18,979 AUDIENCE: [INAUDIBLE] just then having this, I guess, 1701 01:09:18,979 --> 01:09:21,830 easier time with approximate numbers, 1702 01:09:21,830 --> 01:09:24,740 given more of an interest in that. 1703 01:09:24,740 --> 01:09:27,500 NANCY KANWISHER: That's a really, really good question. 1704 01:09:27,500 --> 01:09:29,607 And I am sure there are data on that. 1705 01:09:29,607 --> 01:09:30,899 And I don't know what they are. 1706 01:09:30,899 --> 01:09:32,300 But I will go look. 1707 01:09:32,300 --> 01:09:33,170 I always say that. 1708 01:09:33,170 --> 01:09:36,470 But, Dana, will you send me an email right now to go 1709 01:09:36,470 --> 01:09:39,649 look up whether the prediction from childhood 1710 01:09:39,649 --> 01:09:43,760 ANS to adult arithmetic abilities 1711 01:09:43,760 --> 01:09:45,500 has to do with an interest or you might 1712 01:09:45,500 --> 01:09:48,050 say just an emotional response. 1713 01:09:48,050 --> 01:09:50,600 Like if you suck at it, it feels bad. 1714 01:09:50,600 --> 01:09:52,550 And you become avoidant, and you get all 1715 01:09:52,550 --> 01:09:54,860 dysfunctional about it, right? 1716 01:09:54,860 --> 01:09:59,000 We all have-- I mean, most of us have domains where we do that. 1717 01:09:59,000 --> 01:10:01,400 And math phobia is a real thing. 1718 01:10:01,400 --> 01:10:02,290 And who knows. 1719 01:10:02,290 --> 01:10:03,290 It could start in there. 1720 01:10:03,290 --> 01:10:04,458 So yeah, good question. 1721 01:10:04,458 --> 01:10:05,000 I don't know. 1722 01:10:05,000 --> 01:10:07,230 I will look that up. 1723 01:10:07,230 --> 01:10:10,100 Other questions? 1724 01:10:10,100 --> 01:10:13,870 OK, see you guys on Wednesday.