A Little Psychology of Mathematics

Transcription

1 A Little Psychology of Mathematics

2 Sources What Counts: How Every Brain is Hardwired for Math by Brian Butterworth The Number Sense: How the Mind Creates Mathematics by Stanislas Dehaene

3 Sources The Math Instinct by Keith Devlin Where Mathematics Comes From by George Lakoff and Rafael Núñez

4 The Number Module Brian Butterworth suggests that with rare exceptions we are all born with a Number Module in our brains that categorizes the world in terms of numerosities, that is, the number of things in a collection, and allows us to make use of the cultural tools that extend the capabilities of this number module

5 The Number Module Butterworth s Number Module deals with discrete objects and computes their numerosities. Dehaene, by contrast, suggests that our primary computation circuit is analog, and works more as if it were measuring approximate capacity.

6 The Number Module But both agree that we come hard wired with the capacity to deal with small numbers, and that this skill becomes amplified by social tools we adopt. I ll continue to refer to this by Butterworth s term the Number Module.

7 The Number Module In his book, What Counts, Butterworth provides several pieces of evidence for his theory, and this evidence is worth our looking at because it gives us insight into where mathematics comes from, at least in its simplest forms. We will also examine some of Dehaene s ideas from The Number Sense.

8 Evidences for the Number Module Ability to deal with numerosity in children and animals Widespread ability to deal with numerosity, independent of education Identifiable structures in the brain Fast and automatic operations (comparing or identifying numerosities)

9 Evidences from Animal Studies A chimpanzee named Sheba was trained by Sarah Boysen to understand the meanings of the digits 1 9. Sheba was able to perform arithmetic with them. In one experiment, a number of oranges were hidden in various places in Sheba s cage for example, two oranges under a chair, and 4 more in a box. Sheba s task was to explore her cage, then come back and choose the digit that matched the total number of oranges. She succeeded from the very first trial. Then, rather than oranges, the experimenters hid the digit 2 and the digit 4. Sheba explored the hiding places and then chose the digit 6.

10 Evidences from Animal Studies Two macaques named Able and Baker were able to distinguish the largest among up to five digits. When presented with a string of digits they would use a joystick to point to one of them. They then received that number of fruit candies. They soon learned to pick the largest number. For example, with the sequence they would pick 8 not infallibly, but much more often than chance alone would predict.

11 Evidences from Animal Studies Rats can be trained to press Lever A a set number of times before pressing lever B. A raven name Jakob learned to open the lid of a box having the same number of dots on it as a card alongside two boxes. (2, 3, 4, 5, and 6 spots) A parrot was trained to say the number of objects on a tray. Lionesses can count the number of distinct roars from intruding lions and compare them with the number of friends she has with her.

12 Evidences from Infant Studies Babies can discriminate between 2 and 3 objects a few days after birth. When babies are repeatedly shown cards with two objects on them, eventually they get bored and start spending less and less time looking at them. When a card with 3 objects is then shown, the baby spends more time, demonstrating that it can distinguish between 2 and 3 objects. These are called habituation experiments because they depend on the baby becoming habituated to certain stimuli and then acting surprised (by staring longer) at different stimuli. Infants can also distinguish between words of two and three syllables a few days after birth.

13 Evidences from Infant Studies Four and five month old infants express surprise at impossible arithmetic, such as when two puppets are put behind a screen and the screen drops to reveal either one or three puppets instead. Thus it seems babies know 1+1 = 2, and similarly that 2 1 = 1. Slightly older infants know that = 3 and 3 1 = 2. Babies as young as 6 months can discriminate between 2 and 3 actions, such as a puppet jumping 2 or 3 times.

14 Evidences from Infant Studies In one experiment using six- through eightmonth-olds, babies were presented two stimuli: a card with either two or three common objects on it, and a recording of a drum beating either two or three times. Babies spent more time looking at two-object cards when hearing two beats, and three-object cards when hearing three beats. Thus it seems they could coordinate the numerosity of objects in space and sounds over time.

15 Evidences from Anthropology & Archeology We ve covered this territory Tallies Numeration Systems Counting Systems Five- and ten- and twenty-cycles Counting Boards Number Words Body parts

16 Subitizing Studies have shown that almost universally, humans have the ability to distinguish among 1, 2, and 3 items without counting. This is called subitizing. Most of us can subitize objects up to about four, and we can subitize sequences (or sounds, e.g.) up to about five or so. There is good evidence that this happens suddenly (subitize comes from the Latin word for sudden) and that it is an inborn, universal trait. This may be what the experiments with infants and animals are tapping into.

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21 Naming Time vs Numerosity

22 Subitizing It is interesting to note that most languages have a special status for the number words for one, two, and three. In languages with genders, these first three counting words were the only ones inflected to agree with gender (Old German, zwei, zwo, zween).

23 Subitizing In English, we have the first three ordinal words following a different pattern from the rest: first, second, third, fourth, fifth, sixth,, and on to the twentieth. Finally, the words used for 2 and second often have the connotation of another and words for three often have the connotation of a lot or beyond (e.g. tres in French).

24 Number Module Works well with small collections Allows for distinguishing and comparing numerosities. Also shows interesting effects: Distance effects Compaction of large numbers Stroop Effect Focus on quantity instead of numerical meaning

25 Distance Effects Tell which number in each pair is larger: 2 9

26 Distance Effects Tell which number in each pair is larger: 3 4

27 Distance Effects Tell which number in each pair is larger: 1 8

28 Distance Effects Tell which number in each pair is larger: 5 6

29 Distance Effects Which were easier? 2 and 9 3 and 4 1 and 8 5 and 6

30 Distance Effects Tell whether each of the following pairs are the same

31 Distance Effects 2 9

32 Distance Effects 4 4

33 Distance Effects 3 4

34 Distance Effects 1 8

35 Distance Effects 5 6

36 Distance Effects

37 Compression of Large Numbers It is harder to distinguish between 93 and 97 than between 13 and 17 Distances between numbers are easier to see when the numbers are small, and harder to see when the numbers are large.

38 Compression of Large Numbers Which of these two random series most randomly and evenly samples the interval of numbers between 1 and 2000?

39 Compression of Large Numbers It is actually the first series that covers that interval most evenly:

40 Compression of Large Numbers It is actually the first series that covers that interval most evenly:

41 Compression of Large Numbers

42 Stroop Effect Your job is to name the color of the ink as quickly as you can.

43 Stroop Effect

44 Stroop Effect tuba house ball

45 Stroop Effect Green Orange Green

46 Numerical Stroop Effect Which member of each pair has the largest value?

47 Numerical Stroop Effect 2 9

48 Numerical Stroop Effect 8 4

49 Numerical Stroop Effect 9 8

50 Numerical Stroop Effect 3 8

51 Numerical Stroop Effect Which of the numerals is written in the largest font?

52 Numerical Stroop Effect 2 9

53 Numerical Stroop Effect 8 4

54 Numerical Stroop Effect 9 8

55 Numerical Stroop Effect 3 8

56 Numerical Stroop Effect In both tasks the Name the larger number and the Name the bigger numeral task the answers are quicker if the numerical and physical sizes are coherent. This means that we cannot help but recognize the numerical size quickly enough to have it interfere with the physical size.

57 Attention to numerosity instead of numeral meaning Comparing 79 to 65 is easier than comparing 71 to 65, even though the ten s digits should make both tasks equally easy.

58 How Do We Perceive Numbers? Dehaene suggests we decide on relative sizes of numbers based on a mental number line in which the smaller numbers are closer to us and therefore easier to judge. Larger distances are easier to judge, too. The number line seems somewhat independent of numerical representation. In Western culture, larger numbers are to the right.

59 How Do We Perceive Numbers?

60 A Little About Brains... Double dissociation studies of individuals with certain brain injuries or impairments have shown: The independence of language and number The independence of number and memory The independence of number and reasoning

61 A Side Note Dehaene has suggested that the multiplication tables are stored as verbal information. If this is true, then memorizing them doesn t necessarily make use of their numerical meaning and vice versa. Thus it is something like memorizing the following:

62 The Devil s Address Book Abbey Ernest lives on Zoe Ernest Avenue Bruno Ernest lives on Abbey Zoe Avenue Chloe Ernest lives on Abbey Ernest Avenue Dallas Ernest lives on Bruno Zoe Avenue Ernest Ernest lives on Bruno Ernest Avenue Francis Ernest lives on Chloe Zoe Avenue Gilbert Ernest lives on Chloe Ernest Avenue Henry Ernest lives on Dallas Zoe Avenue Kent Ernest lives on Dallas Ernest Avenue

63 A Little About Brains... Left-hemisphere injuries are more likely to lead to problems with numbers In fact, the left parietal lobe seems to be the most likely candidate for the location of number sense Interestingly, this is where finger manipulation skills are also centered.

64 A Little About Brains... Gerstmann s syndrome has four symptoms: Finger agnosia Acalculia Left-right disorientation Agraphia Butterworth concludes that finger counting provides a major means by which we move beyond the limitations of our Number Module.

65 The Number Module The Number Module is the center of our arithmetic abilities, but they become amplified by cultural tools such as body counting, linguistic representations, and numeration systems. EXTERNAL REPRESENTATIONS CULTURE CONCEPTUAL TOOLS BODY-PART REPRESENTATIONS NUMBER MODULE NUMBER MODULE NUMERALS LINGUISTIC REPRESENTATIONS

66 Back to Philosophy for a Bit For Butterworth, Dehaene, and others, mathematics is a creation of the human brain. The unreasonable effectiveness of mathematics is a result of our brains being evolutionarily programmed to adapt to the world we live in.

67 Back to Philosophy for a Bit Dehaene believes in mathematics by fiat: mathematics models rarely agree exactly with physical reality. Kepler notwithstanding, planets do not draw ellipses... In practice, however, all planets follow chaotic trajectories that merely resemble ellipses and are impossible to calculate precisely beyond a limit of several thousand years.

68 Back to Philosophy for a Bit Dehaene again: Platonism hits upon an undeniable element of truth when it stresses that physical reality is organized according to structures that predate the human mind. However, I would not say that this organization is mathematical in nature. Rather, it is the human brain that translates it into mathematics.

69 Back to Philosophy for a Bit Of course, Dehaene doesn t believe in our God. He sees a universe of chaos and randomness upon which our brains impose order in some cases, mathematical order. Is Platonism the only logical recourse for the believer?