# C82LEA Biology of learning and memory Number. What abilities are involved in numerical competence? 1) Relative numerosity discrimination 2) Absolute.

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1 C82LEA Biology of learning and memory Number

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3 What abilities are involved in numerical competence? 1) Relative numerosity discrimination 2) Absolute number discrimination 3) Ability to count 4) Ability to do arithmetic

4 1) Relative numerosity discrimination The ability to discriminate between sets of items on the basis of the relative number of items that they contain. First to try was Koehler c Emmerton, Lohmann & Niemann 1997 manyfew trained pigeons to discriminate between "few" (1/2 items) and "many" (6/7 items)

5 few many

6 few many.. but are the birds ignoring number, and instead using some other feature of the display? eg light=few dark=many

7 few many.. but are the birds ignoring now DARK=few and LIGHT = many

8 3 4 5 how well do they transfer to new numbers? if they really understand few versus many, they should...

9 1/ /7 novel displays

10 2) Concept of absolute number understanding that 4 bananas and 4 elephants have something in common i.e. number is not intrinsically related to what you are counting

11 Koehler again... Jakob the raven could choose a pot with five spots from an array, even when size of spots varied 50-fold Matsuzawa (1985): chimp called Ai had to select one of six response keys (labelled 1-6) when shown arrays of red pencils, with 1-6 pencils per array. Achieved > 90% accuracy

12 But this is not necessarily the same as counting.... Animals could be learning about specific perceptual pattern -- perceptual matching.

13 But this is not necessarily the same as counting.... Animals could be learning about specific perceptual pattern -- perceptual matching. Argued no, as Ai could transfer her ability to arrays of different types of item 4

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15 the perceptual matching problem... often number is confounded with other factors such as time (for items presented serially) and space (for items presented simultaneously). Are animals using number or these other cues? e.g. smaller number of items also takes up less space. Is it the size of the display controls the response, not number ??

16 with visual arrays there is always going to be something.. so hard to rule out but people have tried in various ways e.g. Pepperberg, 1994

17 Sample of Trials 1. 1 orange chalk, 2 orange wood, 4 purple wood, 5 purple chalk How many purple wood? (4) 2. 1 yellow block, 2 gray block, 4 yellow wool, 6 gray wool How many yellow block? (1) 3. 1 rose wood, 2 blue nail, 3 blue wood, 5 rose nail How many rose nail? (5)

18 Sample of Trials 4. 2 gray truck, 3 gray key, 4 orange key, 5 orange truck How many gray key? (3) 5. 1 blue box, 3 green box, 4 blue cup, 6 green cup How many green cup? (6) 6. 1 purple rock, 2 green rock, 3 purple plastic key, 4 green plastic key How many green rock? (2)

19 another perceptual matching argument... Are the animals subitising? The perception at a glance of the number of items present, without counting them successively; the maximum number of items that can be counted in this way is five

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21 HOW MANY?

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24 The original claim was that subitizing is different from counting because there is little increase in reaction time per item for low numbers of items whereas when dealing with numbers bigger than six, you have to count each one, and because it takes a finite amount of time to count each item the RT increases with number of items This implies that you do not need to count displays of five items or less -- the number is perceived immediately But is this true?

25 However, there is an effect of display size with displays of less than five items -- it takes longer to perceive twoness than oneness, and so on This suggests that even with small displays we are using a counting process

26 Meck and Church (1983): serially presented items. Rats trained with two signals – 2 or 8 pulses of white noise. after 2 were rewarded for left lever response after 8 rewarded for right lever response Respond LEFTRespond RIGHT Each pulse 0.5 sec -- 2 pulse lasted for 2 seconds, 8 pulse for eight seconds. Were animals were responding on the basis of the total time, not number of pulses?

27 To investigate this, they devised a test in which both stimuli lasted 4 seconds: Respond LEFTRespond RIGHT If rats were responding on the basis of stimulus duration, this task should be impossible but they continued to respond correctly

28 To investigate this, they devised a test in which both stimuli lasted 4 seconds: Respond LEFTRespond RIGHT The rats were also tested with pulses of light -- and continued to respond appropriately (Church & Meck, 1984). This is more evidence against perceptual matching Can you think of any other confounds?

29 .. or can make animal respond a fixed number of times – no array involved Davis & Bradford (1991) Access to a plank with food pellets on it Experimenter nearby talking to rat Each rat had designated number of pellets to eat – if he ate more the experimenter shouted No! or clapped loudly. When they ate the right number or fewer than the target they were rewarded by praise and petting (and also a little more food) got it right even when no longer rewarded for correct responses transferred to sunflower seeds -

30 Further evidence from Capaldi & Miller, 1988 Rats trained in a runway, sometimes with food at the end. If the rats expect food they run fast! Trained with following sequences of reinforced (R) trials and nonreinforced (N) trials -- RRRN and NRRRN. Learn to anticipate final N trial and run slow....

31 N R R R N after extensive training.... NRRRN trial... and on an RRRN trial

32 R R R N after extensive training.... NRRRN trial... and on an RRRN trial

33 Learning that three rewards mean no more...? not e.g. length of time in apparatus and were trained with rat pellets; but if one or more of the rewards in the sequence were changed to, for example, cocoa pops, they still did well

34 What abilities are involved in numerical competence? 3) Ability to count Gelman & Gallistel (1978) argued that counting involves mapping numerosity (the property of the display -- e.g. two items) onto a label that represents that numerosity. We usually use number words or symbols as labels, but presumably animals use nonverbal labels, which we can call numerons.

35 The process of counting involves three principles: i) one-to-one principle: each item is assigned only one numeron 14 = 4!! 32

36 The process of counting involves three principles: i) one-to-one principle: each item is assigned only one numeron ii) stable-order principle: numerons must always be assigned in the same order = 4!! 32 = 2!!

37 The process of counting involves three principles: i) one-to-one principle: each item is assigned only one numeron ii) stable-order principle: numerons must always be assigned in the same order iii) cardinal principle: the final numeron assigned applies to the whole display = 4!! 32 = 1!! = 2!!

38 Not just about knowing correct number labels Implies knowledge about order of these labels e.g about how these labels are ordered in relation to quantity e.g. 4>3 2>1 --- ordinal scale and that the size of the difference between each item is the same e.g. 4-3= interval scale

39 Representation of number in the chimpanzee? Biro & Matsuzawa 2000 Ai trained to touch arabic numerals in ascending order

40 Representation of number in the chimpanzee? Biro & Matsuzawa 2000 Ai trained to touch arabic numerals in ascending order But some argued that it was just rote learning of a particular stimulus-response sequence... - no requirement to know anything about the quantitative relation between numbers

41 Representation of number in the chimpanzee? Brannon & Terrace, 2000 Chimps (Benedict, Rosencrantz & MacDuff) trained to order arrays of 1-4 items in ascending, descending, or random order same sizesame surface areavary sizeclip art

42 Representation of number in the chimpanzee? Brannon & Terrace, 2000 Chimps (Benedict, Rosencrantz & MacDuff) trained to order arrays of 1-4 items in ascending, descending, or random order same sizesame surface areavary sizeclip art mixed clip artvary size and shapevary size, shape, colour

43 They could learn ascending and descending orders, but not the arbitrary order Representation of number in the chimpanzee? Brannon & Terrace, 2000 Chimps (Benedict, Rosencrantz & MacDuff) trained to order arrays of 1-4 items in ascending, descending, or random order

44 Representation of number in the chimpanzee? Brannon & Terrace, 2000 Then they were tested with novel displays of 5-9 items 86 75

45 The chimps taught an ascending order could generalize immediately to the higher numbers.... but those taught a descending order could only generalize after further training implies (limited) understanding of the ordering of quantities

46 Alex again... (Pepperberg, 2000) 1 orange chalk, 2 orange wood, 4 purple wood, 5 purple chalk How many purple wood? (4)

47 Alex again...

48 4) Ability to do arithmetic To perform the operations of addition, subtraction etc. To some extent this can be done by rote learning (e.g. times tables); but true mathematical competence would allow these operations to be generalised to new situations in a way that implies a concept of number.

49 4) Ability to do arithmetic To perform the operations of addition, subtraction etc. To some extent this can be done by rote learning (e.g. times tables); but true mathematical competence would allow these operations to be generalised to new situations in a way that implies a concept of number. It is worth asking yourself exactly what this means; is it an all-or- none skill? Or is it a matter of degree? And if the latter, might animals have a limited concept of number?

50 Maths in the chimpanzee? Boysen & Berntson, 1989 A chimp called Sheba was trained to label arrays with counters, and then with arabic numerals:

51 Maths in the chimpanzee? Boysen & Berntson, 1989 A chimp called Sheba was trained to label arrays with counters, and then with arabic numerals:

52 Maths in the chimpanzee? Boysen & Berntson, 1989 A chimp called Sheba was trained to label arrays with counters, and then with arabic numerals:

53 Maths in the chimpanzee? Boysen & Berntson, and then with arabic numerals:

54 Maths in the chimpanzee? Boysen & Berntson, 1989 She also performed well when items swapped for everyday objects

55 She was given extensive training with numbers 0-4

56 In the final test a number of oranges were hidden in the lab, in any of three hiding places. Sheba had to find all the oranges, and then pick the arabic numeral that represented the sum of all the oranges that were hidden. After 12 training sessions (of around 20 trials per session) she was performing at about 85% correct. Answer = 3

57 Potential problems..... you could argue she memorized all the ways of adding 0,1,2,3,4 to a total of but....!

58 She could also perform accurately when the experimenters hid cards with numbers written on them, rather than oranges -- and she performed above chance right away implies understanding of the interval scale – if she understood only bigger than she would have chosen 4 as often as 3 1 Answer = 3 2

59 In another experiment ( Boysen & Bertson,1995) chimp A was given a choice between two amounts of candy. Whichever chimp A chose was given to a second chimp, B, and chimp A got to eat the other one.

60 A chooses B A In another experiment ( Boysen & Bertson,1995) chimp A was given a choice between two amounts of candy. Whichever chimp A chose was given to a second chimp, B, and chimp A got to eat the other one. It was thus in chimp As interest to choose the smaller quantity, so it could eat the larger quantity. They were completely unable to solve this task -- unless the candy was substituted by numerals.

61 A chooses B B A A 1 3 In another experiment ( Boysen & Bertson,1995) chimp A was given a choice between two amounts of candy. Whichever chimp A chose was given to a second chimp, B, and chimp A got to eat the other one. It was thus in chimp As interest to choose the smaller quantity, so it could eat the larger quantity. They were completely unable to solve this task -- unless the candy was substituted by numerals!

62 A chooses B B A A 1 3 Is this evidence they can't count? Or just that they can't resist a treat.. correct motivation critical for good performance

63 General references Pearce, J.M. (1997). Animal Learning and Cognition. Lawrence Erlbaum Associates. Chapter 7. Shettleworth, S.J. (1998). Cognition, Evolution and Behaviour. Oxford University Press. Chapter 8 and pp Wynne, C.D.L. (2001). Animal Cognition. Macmillan. Chapter 5 pp Specific references Biro, D., & Matsuzawa, T. (2000). Numerical ordering in a Chimpanzee: Planning, executing and monitoring. Boysen S.T., & Berntson, G.G. (1989). Numerical competence in a chimpanzee. Journal of Comparative Psychology, 103, Boysen S.T., & Berntson, G.G. (1995). Responses to quantity: perceptual versus cognitive mechanisms in chimpanzees. Journal of Experimental Psychology: Animal Behavior Processes, 21,

64 Brannon, E.M., & Terrace, H.S. (2000). Representation of the numerosities 1-9 by rhesus macaques. Journal of Experimental Psychology: Animal Behavior Processes, 26, Capaldi, E.J., & Miller, D.J. (1988). Counting in rats: Its functional significance and the independent cognitive processes that constitute it. Journal of Experimental Psychology: Animal Behavior Processes, 14, Church, R.M., & Meck, W.H. (1984). The numerical attribute of stimuli. (pp ) In Roitblat, H.L., Bever, T.G., & Terrace, H.S. (Eds.) Animal Cognition. Lawrence Erlbaum Associates. Davis, H, Bradford, S.A. (1991) Numerically restricted food intake in the rat in a free- feeding situation. Animal Learning & Behavior, 19, Emmerton, J, Lohmann, A., & Niemann J. (1997). Pigeons' serial ordering of numerosity with visual arrays. Animal Learning & Behavior, 25, Gelman, R., & Gallistel, C.R. (1978). The childs understanding of number. Cambridge, MA: Harvard University Press. Matsuzawa, T. (1985). Use of numbers by a chimpanzee. Nature, 315,

65 Meck, W.H., & Church, R.M. (1983). A mode control model of counting and timing processes. Journal of Experimental Psychology: Animal Behavior Processes, 9, Pepperberg, I.M. (1994). Numerical competence in an african gray parrot. Journal of Comparative Psychology, 108, Pepperberg, I.M. (2000). Ordinality and inferential abilities of a grey parrot. Journal of Comparative Psychology, 120,

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