Numbers and Counting Flashcards
what abilities are involved in numerical discrimination?
- Relative numerosity discrimination (many vs a few)
- Absolute number discrimination (4 vs 5)
- Ability to count (1,2,3,4)
- Ability to do arithmetic (1 + 2 =)
what is relative numerosity discrimination?
- the ability to discriminate between sets of items on the basis of the relative number of items that they contain
- the first to try this was Koehler c. 1913
Relative numerosity discrimination study, Emmerton et al., 1997
- trained pigeons to discriminate between “few” (1/2 items) and “many” (6/7 items)
- could be possible that birds are ignoring number, and instead using some other feature of the display e.g. LIGHT = few, DARK = many
what is concept of absolute number?
understanding that despite differing appearances, 4 bananas and 4 elephants have something in common
i.e. number is not intrinsically related to what you are counting it’s an abstract thing unrelated to physical characteristics
concept of absolute number study, Koehler and Matsuzawa (1985)
- Jakob the raven could choose a pot with 5 spots from an array, even when the size of spots varied 50-fold
concept of absolute number study, 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.
limitation of concept of absolute certainty
- But this is not necessarily the same as counting
- Animals could be learning about specific perceptual pattern called perceptual matching
- arrays of e.g. four objects have more in common with each other
than arrays of e.g. 2 or 7 - Matsuzawa argued no: Ai could transfer her ability to arrays of different types of item
Serially presented numbers, Meck and Church (1983)
- 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
- each pulse 0.5 sec - ‘2 pulse’ lasted for 2 seconds, ‘8 pulse’ for eight seconds
evidence against perceptual matching, Meck and Church (1983 and 1984)
- animals could be responding on the basis of the total time, rather than number of pulses
- so devised test in which both stimuli lasted 4 seconds
- if rats responded on the basis of stimuli duration, this task should be impossible, but they responded correctly
- the rats were also tested with pulses of light and continued to respond appropriately
- this is more evidence against perceptual matching
Can you make animals respond a fixed number of times, David and Bradford, 1986
- rats had access to a plank with food pellets
- experimenter stood nearby talking to the rat
- each rat had a designated number of food pellets to eat
- if the rat ate more than the desired amount the experimenter shouted “No!” loudly or clapped
- when they ate the right number or fewer than they target number they were rewarded by praise and petting and a little bit more food
- found that the rats got it right even when no longer rewarded for correct responses
- this was also found with sunflower seeds
Ability to count, Gelman and Gallistel (1978)
- they argued that counting involving mapping numerosity (the number in the display, e.g. 2 items) onto a label that represents that numerosity
- we usually use number words (one, two) or symbols (1,2) as labels, but animals must use nonverbal labels - these are called numerons
what 3 principles of the process in counting?
- one-to-one principle: each item is assigned only one numeron
- stable-order principle: numerons must always be assigned in the same order
- cardinal principle: the final numeron assigned applies to the whole display
what does knowledge of the correct number labels imply?
- implies knowledge about order of these labels
- about how these labels are ordered in relation to quantity e.g. 4>3 (ordinal scale)
- and that size of the difference between each item is the same e.g. 4-3=3-2 (interval scale)
Representation of number in chimpanzees, Brannon and Terrace (2000)
- chimps were trained to order arrays 1-4 items in ascending, descending or a random order
- they could learn the ascending and descending orders, but not the arbitrary order (1,2,3,4 compared to 1,3,2,4)
then they were tested with novel displays of 5-9 items - the chimp was taught an ascending order could generalise immediately to the higher numbers
- but the one taught a descending order needed further training
- implies (limited) understanding of the ordering of quantities
what does it mean by the ability to do arithmetic?
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
