An Example of a "Learning Process" Journal (using the 2 colored box format)

Book: The Number Sense by Stanislas Debaene

Source: Oxford University Press 1997

Yellow highlighted passages: Introduction

• "Every single thought we entertain, every calculation we perform, results from the activation of specialized neuronal circuits implanted in our cerebral cortex." [p. 4]
• "Even our digital notation of numbers ... is the fruit of a slow process of invention over thousands of years." [p. 4]
• "The slow cultural evolution of mathematical objects is a product of a very special biological organ, the brain, that itself represents the outcome of an even slower biological evolution governed by the principles of natural selection." [p. 4]
• "All people possess, even within their first year of life, a well-developed intuition about numbers." [p. 5]
• "Humans, however, have been endowed by evolution with a supplementary competence: the ability to create complex symbol systems, including spoken and written language. Words, or symbols, because they can separate concepts with arbitrarily close meanings, allow us to move beyond the limits of approximation." [p. 5]
• "The concept of number, hinted at by the Babylonians, refined by the greeks, purified by the Indians and the Arabs, axiomitized by Dedekind and Peano, generalized by Galois, has never ceased to evolve from culture to culture - obviously, without requiring any modification of the mathematican' genetic material! [p. 6]
• "Our brain ... has been endowed since time immemorial with an intuitive representation of quantities. ... Our memory ... is not digital but works by association of ideas. This is probably the reason why we have such a hard time remembering the ... multiplication table." [p. 7]
• "Mathematicians have worked hard for centuries to improve the usefulness of numerical notations by increasing their genrality, their fields of application, and theri formal simplicity." [p. 7]

The first quote should never be forgot!

We should be both amazed and humbled by the history of mathematics. Why did it take so long for us to arrive at our present notational conventions? But look how impressive our accomplishments in mathematics have been!

Humans are fundamentally analog machines, but our use of symbols permits us to also be digital. We are bilingual!

Yellow highlighted passages: Chap. 1 Talented and Gifted Animals

• "In nuerous species, estimating the number and ferocity of predatores, quatifying and comparing the return of two sources of food, are matters of life and death." [p. 14]
• "The ability to compare two numerical quantities ... is widespread among animals." [p. 25]
• "... the nature of these errors provides important cues about the nature of the mental representation employed. ... they demonstrate ... that animals do not possess a digital or discrete representation of numbers." [p. 27]
• "To a rat, numbers are just approximate magnitudes." [p. 30]
• "When we count, we use a precise sequence of number words. ... Not so for rats. ... The difference with our verbal counting is so enormous that we should perhaps not talk about 'number' in animals at all ...'numerosity' ... enables animals to estimate how numerous some events are, but does not allow them to compute their exact number." [p. 34 - 35]
• "While we may marvel at animals' ability to manipulate approximate representations of numerical quantities, teaching them a symbolic language seems to go against their natural proclivities." [p. 39]
• "... if our closest cousins, the chimpanzees possess some competence for arithmetic, and if species as different as rats, pigeons, and dolphins are not devoid of numerical abilities, it is likely that we Homo sapiens have received a similar heritage. Our brains, like the rat's, are likely to come equipped with an accumulator that enables us to perceive, memorize, and compare numerical magnitudes." [p. 40]
• Many outstanding differences separate human cognitive abilities from those of other animals:
  • we have an uncanny ability to develop symbol systems ...
  • we are also endowed with a cerebral language organ that enables us to express our thoughts and to share them with other members of our species ...
  • our ability to devise intricate plans for actions, based on a retrospective memory of past events and a prospective memory of future possibilities, seems to be unique in the animal kingdom." [p. 40]

The basic point is that we are more than simply analog numerical processors, using language and symbol systems to develop new ideas.

Yellow highlighted passages: Chap. 2 Babies Who Count

• "Yet if our working hypothesis is correct, the human brain is endowed with an innate mechanism for apprehending numerical quantities, one that is inherited from our evolutionary past and that guides the acquisition of mathematics. To influence the learning of number words, this protonumerical module must be in place before the period of exuberant language growth that some psychologists call the 'lexical explosion', which occurs around a year and half of age." [p. 41]
• "...replacing marbles with palatable treats (M&Ms) ... the children were allowed to pick up one of the two rows and consume it right away. ... a majority of children selected the larger of the two numbers, even when the length of the rows conflicted with number." [p. 45]
• "That three- and four-year-old children select the more numerous row of candy ... conflicts directly with Piaget's theory. But there is more. ... the youngest children, who were about two years old, succeeded perfectly in the test, both with marbles and with M&Ms. Only the older children failed to conserve the number of marbles. ... For some reason, these tests seem to confuse the older children." [p. 45]
• "I believe that what happens is this: Three- and four-year olds interpret the experimenter's quesions quite differently from adults. The wording of the questions and the context in which they are posed mislead children into believing that they are asked to judge the length of the rows rather than their numerosity. Remember that, in Piaget's seminal experiment, the experimenter asks the very same question twice; 'Is it the same thing, or does one row have more marbles?' He first raises this question when two rows are in perfect one-to-one correspondence, and then again after their length has been modified. What might children think of these two successive questions? Let us suppose that the numerical equality of the two rows is obvious to them. They must find it quite strange that a grown-up would repeat the same trivial question twice. Indeed, it constitutes a violation of ordinary rules of conversation to ask a question whose answer is already known by both speakers. ... 'If these grown-ups ask me the same question twice, it must be because they are expecting a different answer. ..." [p.45-46]
• "Piagetian conservation tasks ... this is an active domain of research that still attracts many researchers throughout the world." [p. 47]
• "... we now know what Piagetian tests are not about. ... these are not good tests of when a child begins to understand the concept of number." [p. 47]
• "The experiments that I have described so far challenge the Piagetian time scale for numerical development by suggesting that children 'conserve number' at a much earlier age than was once thought possible." [p. 47-48]
• "... even newborns coud discriminate numbers 2 and 3 a few days after birth." [p. 50]
• "... use a sucking rhythm rather than a gaze orientation ..." [p. 51]
• "While young children's numerical abilities are real, they are strictly limited to the most elementary of arithmetic." [p. 56]
• "... only occassionally are they shown to differentiate three versus four. And never can a group of babies under one year of age distinquish four dots from five or even from six." [p. 57]
• "Babies, however, ... most likely possess only an approximate and continuous mental representation of numbers." [p. 57]
• "Newborns readily distinquish two objects from three ... their brain apparently comes equipped with numerical detectors that are probably laid down before birth." [p. 61]
• "Younger babies seem unaware of the natural ordering of numbers." [p. 63]

I was not aware of the use of a sucking action as a substitute for watching their eyes.

I also was not aware that 2 year-olds could "conserve number" but that three-year-olds had difficulty.

I agree with the conclusion about pre-wiring, that it only refers to very small numbers, and that it does not imply any abstract sense of number or the relationship among numbers. It is simply a primitive sense of which is different.

Psychology Index