1. Studies on Number Concepts Psych 1090 Lecture 5

2. We looked at number concepts at the beginning of the course … Now we ’ ll look at them in quite a bit more details Now from the standpoint of trying to figure out exactly what types of number concepts animals have

3. What constitutes numerical competence? Number can be a descriptive category  chose one set from competing arrays  chose with respect to “more”/”less”  match-to-sample with respect to quantity  respond to one sequential series of events Most of these do not involve exact number….

4. Even if subjects discriminate a specific amount, such as ‘ threeness ” when given a variety of choices

5. The subjects  may have only a recognition of pattern  may actually be “subitizing”….

6. Subitizing, as we learned earlier,  is a perceptual mechanism  generally depends on canonical arrays -- think dice, dominoes  is used when time constraints exist  is usually approximate for larger quantities -- but can be sequential

7. If a subject is choosing “ more ” versus “ less ” Unless the items vary considerably

8. Subjects may be responding to  mass or contour  brightness or density

9. In fact, some work on children (Spelke, Brannon) has shown exactly that …. when the samples were controlled for mass and contour, the purported numerical sensitivity disappeared

10. What might you prefer? Three small bites, or one huge piece?

11. In terms of match-to-sample Again, unless object differ ….

12. Various mechanisms having very little to do necessarily with number can kick in And even if the objects do differ somewhat Pattern recognition can be used to solve the problem

13. And, as we mentioned in lecture 2 Even if the patterns are sequential,

14. Most experiments just examined responding to differences Such as two versus four …. Whereby the subjects could again respond only with respect to “ more ” versus “ less ”

15. Numerical research in animals has a long history…. Current thinking suggests that humans and animals share processes that involve subitizing…. Various proposals exist for the mechanisms involved….e.g., object files, accumulators….

16. And we need to spend a bit of time differentiating these mechanisms So that we can understand how they differ from true counting

17. One study to claim that children recognize exact numbers — and that actually claimed addition and subtraction Involved a looking-time paradigm

18. The idea was that children become habituated to a set And then respond only if there is a change in the set so they are shown something like this ….

20. Until they get bored and stop looking …

21. Then they are shown

22. To see if they are startled by the change

23. The initial studies with children did not test for  size or mass differences  a change in the object  pattern (i.e., sideways figures)

24. The argument was that the only thing changed was number and hence the children (at 5 mos) had to be responding to a difference in number but the infants might simply have realized something changed without knowing WHAT

25. In fact, some researchers in the UK have done some brain studies that suggest that it really is just boredom And then some attention mechanism that is independent of the type of change being made

26. So the mechanism wasn ’ t necessarily counting or even number sensitivity And we ’ ll discuss the related animal study in a bit, Which did, of course, add some controls

27. Another set of ideas suggests that subjects without language be they human or nonhuman use one noncounting mechanism for small groups,  4 And then estimate larger quantities

28. The big question is then what kind of mechanism is used for the smaller amounts Two main competitors exist:  Accumulator models  Object file models

29. The accumulator model would seem to work very well for sequential tasks … The model, set up to explain both counting and timing, was first presented by Mech and Church

30. Here there is a sort of gating mechanism in which the brain tags each event as it ‘ fills up ’ or ‘ clicks by ’…

31. The mechanism works very well for small numbers Usually less than four After which it becomes approximate

32. So that, for example, if the number of events is six You get a curve with the center about 6 And with tails on either end

33. percentage 0 1 2 3 4 5 6 7 8 9 10 11 12 60

34. The system doesn ’ t work as well for simultaneous arrays Because it would have to posit that each object in the array was being scanned at a constant rate

35. So researchers suggested an “ object file ” in which the subject hold a representation of the items as a sort of pattern array in short-term memory

36. Somewhat like lambs gamboling around in a meadow So that about the maximum that can be kept in short- term memory is about 4 And these are compared with a mental memory of various sets

37. This, too, is an approximate method And the results for larger numbers look like the same curve we saw before Its advantage is that it does not require sequential presentation

38. The difference between these mechanisms has to do with what the differences for curves look like for a given number …. For larger numbers on the accumulator model we expect more errors and more severe errors as the numbers increase

39. For the object file, the system just breaks down for the larger sets and a general estimation process takes over

40. Now remember, these are not supposed to be counting mechanisms But rather something different from the perceptual system of subitizing Which really does need some kind of pattern array for larger numbers

41. So let ’ s look at the monkey paper first … We saw some of the video in lecture 2 for some of the sets of trials And the journal article discusses the others …

42. Note in the studies on monkeys prior to this one that monkeys failed at discriminating large sets So if they saw three apples being put into one box

43. and eight apples being put into another they chose the boxes at chance suggesting that after about 4, they simply couldn ’ t keep track of what was happening even with respect to sequential ‘ more ’ versus ‘ less ’

44. So, the first experiment in the paper we read …

45. As in the earlier experiments The animals simply reacted to some level of difference Although with respect to a slightly larger system

46. The second experiment tested whether the monkeys were responding to the amount of ‘ stuff ’ Note that monkeys had previously chose number over mass if the total mass was the same …. But that didn ’ t track contour

47. Of course, this again marks difference in general ….

48. If you saw lots of little things disappear Wouldn ’ t you be surprised if one big thing appeared? Unless you were looking a stuff like piles of sand?

49. In introducing the next experiment the researchers first describe an experiment with children Stating that the children discriminate 3-1=2 versus 2+1=2

50. But let ’ s think about this … so you expect some change

51. And here you also expect change

52. So the reaction is to the expectation of change of some sort But again, not necessarily with respect to number, just pattern

53. The interesting issue was why older children didn ’ t discriminate 3-1=3 versus 3-0=3 issue shouldn ’ t just be too much stuff … Maybe the order of testing?

54. Now, on this next experiment It couldn ’ t just be change ….

55. Because then the monkeys would be perfectly happy with 4 So is this some actual evidence for number? Better, but it still could be mass or contour expectation

56. In the next experiment The monkeys had to update their information twice In addition to having to deal with the larger numbers

57. and this time they fell apart completely

58. So now, the question was whether it really was the number of updates that blew the task for the monkeys … This experiment has to do with separating out object files and accumulators ….

59. If the monkey used an accumulator for a 1+1+1=3 versus 1+1+1=2 it should succeed, because the values are still quite small But if it failed, on a quantity it could do with a single update

60. That might suggest object files …

61. Remember, in reality, neither object files nor accumulators require actual counting But rather some attention to ‘ stuff ’ for the former and actions for the latter … And both are approximations …

62. The argument for object files holds because the animals have to keep a running total as they make a comparison with a mental representation And that ’ s more difficult

63. So their failure on the sequences suggest that they are using object files But, again, nothing here really argues for counting

64. Now, researchers like Brannon have looked at monkeys ’ abilities with far greater numbers … And claim that there exists a basic, nonverbal number concept across all primates at least in terms of comparisons …

65. One of the arguments is that discrimination between two numbers has to do with their ratios Obviously, it ’ s easier to see the difference between 3:8 versus 7:8 in terms of real ‘ stuff ’ And also with the Arabic numerals if they represent real ‘ stuff ’

66. Of course, one really finds that kind of difference only when there ’ s a time constraint …. We ’ ll see that there isn ’ t that much of a difference for Alex But for almost all his tests, he has unlimited time to make his discriminations

67. Now, remember that it took the monkeys about 50,000 trials to learn to make the orderings of 1 through 9 …. Which suggests that the concept was not one that was easily acquired … But that is less important than transfer …

68. So the monkeys saw arrays like this And had to choose the one with fewer things first

69. With smallish sets or big ratios, it ’ s not that tough … But you could imagine how difficult it would be for, say, 8 vs 9 blobs.. The data reflect these issues very nicely …

70. Overall (including lots of easy trials) the monkeys did quite well …

71. But, as you might expect, the monkeys were at chance when the ratios were very close … As were humans

72. Brannon thus argues that the mechanisms are the same … Some us of ‘ mental magnitudes ’… I don ’ t disagree at all with the results or the argument that basic processes are at work … I do not, however, think that mental representations exist in the monkeys

73. Unless 50,000 trials have sensitized the animals …. But one still has to put this into the perspective of the Carey and Hauser work on sequential number …. Was it the ratios in C&H that were the problem?

74. If so, then we are looking at distinct mechanisms for more versus less and actual number recognition …. which is not at all surprising …. and ties into idea on counting ….

75. “Counting” is a very specific behavior:  Produce a standard sequence of number tags (but maybe not out loud)  Apply a unique tag to each item to be counted  know that the last number tag used tells the quantity of interest

76. Most researchers argue that true counting can exist only with language ….even for humans… And, of course, most animals do not have language…. However, a few apes, dolphins, and parrots have acquired elements of human communication systems, including number labels…..

77. So, let’s start talking about creatures who have been trained to label quantities

78. Matsuzawa ’ s Ai learned to associate up to 10 blobs with the Arabic numeral 10 Like humans, she was fast and accurate on 1-4 Presumably subitizing

79. And, like humans for 5-9, She took longer for each addition item She was also less accurate, which isn ’ t really true for adult humans (unless time is restricted) but can be true for children

80. Interestingly, Ai (and also Sally ’ s chimps) did not show much savings for learning the larger numbers in order Such data suggest that they hadn ’ t really set up a representation of number as a sequence

81. In fact, Ai seemed to do best on the largest number as compared to the next largest number Which suggested some idea of using the last number as “ many ” But also confusing that with the next smaller one..

82. Now, we don ’ t know if this was exactly true for Alex …. First, we didn ’ t teach him numbers in order … we did 3 and 4 at the same time, then 5, then 2 then 6 then 1 Specifically because we didn ’ t want him to have the advantage of knowing that each symbol was “ one more ”

83. I didn ’ t want that to be some kind of cue, but wanted the symbol to be a total representation Second, Alex wasn ’ t simply associating a visual symbol with an array but rather talking …

84. Which meant that he had to learn how to get all his articulatory muscles and air patterns into a specific configuration …. And think about “ f ” and “ v ” without lips ….

85. But in all cases, the question was whether the nonhuman was actually counting, or doing something else And, in most cases, the only way to tell was to do a timed experiment …. If an animal could use a perceptual strategy for large quantities

86. Its accuracy, unlike that of humans, Would not decrease as the amount of time allowed to do the task decreased Which is what Matsuzawa tested

87. So one screen had a random array of dots And, very importantly, the other screen had a random array of numerals So that there couldn ’ t be just a 1:1 association of symbols But the symbols had to have meaning

88. What I did find odd was that not every numeral was present on every trial At least according to the figure presented in the paper …

89. I don ’ t see a 2 in either one, or an 8 or 9 on the right …

90. The data are quite interesting

91. Unlike humans, Ai ’ s errors on all numbers other than 9 didn ’ t differ much between brief exposure and unlimited exposure Humans erred less overall, but particularly on 9 And response times were really telling

92. Ai really seemed to take lots of time when it was available to distinguish 8 from 7 and 9 But seemed to hit 9 as the upper limit, as tho ’ it were ‘ lots ’ And she did more looking back for larger numbers

93. Not surprisingly, Ai had trouble initially distinguishing some symbols ….1-7, 6-9, 3-8, 2-4 Alex had some of the same issues til we used very distinct 7s, and we do not plan to use 9 But he confused 2 with 5, which makes sense, too

94. The researchers argue that Ai did not count for the brief times but rather estimated The humans could have clumped … quickly saw patterns of small sets and added the sets What Ai was doing may be something similar

95. The stability of Ai ’ s response time during brief exposure suggests that she did some form of estimation She would have been more accurate had she been able to subitize Counting would have led to her using all the time possible and high error rates for 5-8

96. But, of course, if she were clumping, that process would be consistent with no real time change for 5-8 It would be really exciting to see how Ai did on the task given Alex to separate out processes …

97. Remember, in other studies when humans weren ’ t given enough time

102. Researchers found that humans bottomed out at about the same level as the pigeons–- about 4

103. And that Trick and Pylyshyn went about this in another way showing confounded sets of colors and objects as well as numbers

105. Number used by Alex Number of objects 1 2 3 4 5 6 1 2 3 4 5 6 7 8 6 9 7 1 2 1 1 11 4 1 1 8

106. But we also have to look at Boysen ’ s chimps Who also were trained with human symbols Much like Ai

107. Initially, Sheba learned to touch the set of numbers and then the Arabic number that corresponded to a small array of candies

108. She was then trained on comprehension before going on to larger numbers This procedure was unlike that given Alex, who was not given any training on comprehension Comprehension is critical …

109. Children who can look at an array and tell you that there are “ four ” marbles …. But if you give them a big bowl and ask for “ four ” Young children will just grab a bunch …

110. Boysen ’ s Sheba initially had trouble when four was introduced; The table presented does not show how long it took in terms of trials per day But Sheba eventually did well on both “ 0 ” and “ 4 ”

111. Sheba was then tested on production using random ‘ junk ’ items in the lab But now the Arabic numbers were always placed in ordinal sequence …. Thus Sheba could be working on numbers of cards ….

112. As we mentioned in lecture 2, I had to check out Alex’s comprehension But I wanted to be extremely careful that he was responding only with respect to the objects and nothing else

114. Number Alex produces Number of objects 1 2 3 4 5 6 n ? 1 2 3 4 5 6 8 9 10 8 8 1 1 1 1 11 2 none 5 1

115. Alex’s results demonstrate competence comparable to that of nonhuman primates and young children And we have previously discussed his spontaneous use of “none” to represent absence of quantity

116. But what about something like addition ? Boysen’s chimps could walk around the room, look at different collections, then point to the Arabic number that represented the sum Sally argues that it is counting

117. The reason being that Sheba had to remember the representations as she went around the room …. But it is possible that, particularly given the small numbers involved in this study That Sheba was clumping the arrays

118. That ability, in and of itself, is still pretty exciting … And in later work, Boysen extended the material up to about 6 (and maybe 8, tho ’ that is not in peer-reviewed journals)

119. And we began a study to replicate the Boysen & Berntson work…

120. Number Alex produces Number of objects 1 2 3 4 5 6 n ? 1 2 3 4 5 6 8 7 8 7 4 11 7 1 2 none 1 11 3 5

121. Remember, Alex did replicate Sheba ’ s abilities, But remember that Sally gave Sheba as much time as needed And we initially time-limited Alex … And he couldn ’ t do 5+0 …

122. Data were for first trials only, and he was correct only 50%… When he errs, he gets a total of 4 chances…and almost every time he said “6”….

123. Was he engaging in a counting-like strategy for “5”?? Doesn’t seem to be using an accumulator or object files or subitizing, because he’s too accurate If we gave him 10 instead of 2 s, he was 100% accurate… Was he subitizing  4, and seeing 6 as “lots”?

124. It would also be interesting to see what Sheba would have done in a time-limited situation, like that of Alex or Ai But Sheba did something else that was extremely important

125. Whether or not she was counting the objects or just summing them in some manner Boysen replaced the objects with the Arabic numerals And replicated the study

126. Even on the first few trials Sheba was above chance The critical test for Sheba was not 0+n Which was just choose “ n ”

127. But 2+2, because just avoiding addend wouldn ’ t work Nor would choosing the next number in the series But she had to have a representation

128. What would be interesting for both Alex and Sheba would be to have to update their sets more times … Like the monkeys …. What would they do with 2+2+2?

129. Now, other work on numbers on chimps and parrots involve their full understanding of ordinality Do they really understand that their symbols represent a linear order …. that 2 1?

130. As it turns out, Boysen had to train her chimps; Even Sheba, who had done addition needed about 500 trials to get all the ordering done correctly

131. We did this with Alex, too …. First, we trained him to identify Arabic numbers by telling us their colors So, he saw a tray like this …

132. 1 6 4 5 2 3

133. The task was not simple, and it took awhile Partly, I believe, because of something called “ mutual exclusivity ”…. If six blobs were “ six ”, why was one squiggle also “ six ” ?

134. But eventually he got to respond above chance And he had previously learned how to respond to “ What color bigger/smaller? ” for two objects

135. So now we showed him two differently colored Arabic numbers …

136. And he was correct at a statistically significant rate We also asked him about the same Arabic numerals, and he said “ none ” And used numbers that themselves were bigger or smaller

137. Too much data to discuss here But basically he was able to do all the various combinations Including sets with one Arabic number and a collection of items The take-home messages:

138. Given that parrots and primates evolutionary history dates from the dinosaurs…. Number concepts are likely to be relatively widespread across species And we haven ’ t talked about other critters …

139. Maybe numerical competence involves giving the subject the appropriate tools to express latent abilities…. Certainly, enculturation is important, given evidence from untrained humans in Peru

140. Future directions for number work---  training larger numbers  determining if Alex will comprehend new number labels more quickly than old ones  completing subtraction studies  sequential sounds in younger birds and transfer to simultaneous visual

141. But I hope what came through was that the critical issue in determining these abilities has to do with experimental design

142. Sci-Am show on number work in chimps