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: Chap. 5 Small Heads for Big Calculations

• "While children easily acquire number syntax, learning to calculate can be an ordeal. ... Why is mental calculation so difficult?" [p. 118]
• "An innate sense of approximate numerical quantities may well be embedded in our genes; but when faced with exact symbolic calculation, we lack proper resources." [p. 119]
• "By their fourth year, children have mastered the basics of how to count. ... The origins of this precocious competence remain poorly understood." [p. 120]
• "Reciting words in a fixed order is probably a natural outcome of the human faculty for language. As to the principle of one-to-one correspondence, it is actually widespread in the animal kingdom. ... The counting algorithm stands at the intersection of these two elementary abilities of the human brain - word recitation and exhaustive search." [p. 120]
• "Though children rapidly grasp the how to of counting, however, they seem to initially ignore the why. ... children do not appreciate the meaning of counting until the end of their fourth year." [p. 121]
• "Remember that right from birth, way before they start to count, children have an internal accumulator that informs them of the approximate number of things that surround them. ... Suppose the child is playing with two dolls ... the child has learned that the word 'two' applies to this quantity [subitizing plus vocabulary, just like learning when to use the word dog], so that he can say 'two dolls' without having to count. Now suppose that for no particular reason, he decides to 'play the counting game' with the dolls, and recites the words 'one two.' He will be surprised to discover that the last number of the count, 'two', is the very word that can apply to the entire set. After ten or twenty such occasions, he may soundly infer that whenever one counts, the last word arrived at has a special status. ... Counting, which was only an entertaining word game, suddenly acquires a special meaning: Counting is the best way of saying how many!" [p. 121]
• "With the help of counting, most children find ways of adding and subtracting numbers without requiring any explicit teaching." [p. 122]
• "Calculation abilities do not emerge in an immutable order. Each child behaves like a cook's apprentice who tries a random recipe, evaluates the quality of the result, and decides whether or not to proceed in this direction." [p. 121]
• "During the preschool years ... Children suddenly shift from an intuitive understanding of numerical quantities, supported by simple counting strategies, to a rote learning of arithmetic. ... this major turn coincides with the first serious difficulties that children encounter in mathematics." [p. 126]
• "What makes the multiplication table so much harder to retain, even after years of training? The answer lies in the particular structure of the addition and multiplication tables." [p. 127]
• "If our brain fails to retain arithmetic facts, that is because the organization of human memory, unlike that of a computer, is associative: It weaves multiple links among disparate data, Associative links permit the reconstruction of memories on the basis of fragmented information." [p. 127]
• "Associative memory is a strength as well as a weakness. It is a strength when it enables us, starting from a vague reminiscence, to unwind a whole ball of memories that once seemed lost. ... It is a strength again when it permits us to take advantage of analogies and allows us to apply knowledge acquired under other circumstances to a novel situation. Associative memory is a weakness, however, in domains such as the multiplication table where the various pieces of knowledge must be kept form interfering with each other at all costs. ... But when trying to retrieve the result of 7x6, we court disaster by activating our knowledge of 7+6 or of 7x5. ... our brain evolved for millions of years in an environment where the advantages of associative memory largely compensated for its drawbacks." [p. 128]
• "In order to calculate fast, the brain is forced to ignore the meaning of the computations it performs." [p. 132]
• "Clearly, the human brain ... has not evolved for the purpose of formal calculation." [p. 134]
• "We cannot hope to alter the architecture of our brain, but we can perhaps adapt our teaching methods to the constraints of our biology. Since arithmetic tables and calculation algorithms are, in a way, counter natural, I believe that we should seriously ponder the necessity of inculcating them in our children." [p. 134]
• "I am convince that by releasing children from the tedious and mechanical constraints on calculation, the calculator can help them to concentrate on meaning." [p. 135]
• "Electronic calculators, as well as mathematical software for children, hold the promise of initiating them to the beauty of mathematics: a role that teachers, all too occupied in teaching the mechanics of calculation, often do not accomplish." [p. 136]
• "Yet it should be acknowledged that today the vast majority of adults never perform a multidigit calculation without resorting to electronics. Whether we like it or not, division and subtraction algorithms are endangered species quickly disappearing from our everyday lives - except in schools, where we still tolerate their quiet oppression." [p. 136]
• "In China, children are explicitly taught to reorder multiplications by placing the smallest digit first. This elementary trick, which avoids relearning 9x6 when one already knows 6x9, cuts the amount of information to be learned by almost one half." [p. 136]
• "Lacking any deep understanding of arithmetic principles, they are at risk of becoming little calculating machines that compute but do not think." [p. 136]
• "... number knowledge does not rest on a single specialized brain area, but on vast distributed networks of neurons, each performing its own simple, automated, and independent computation." [p. 138]
• "... schooling plays a crucial role not so much because it teaches children new arithmetic techniques, but also because it helps them draw links between the mechanics of calculation and its meaning. A good teacher is an alchemist who gives a fundamentally modular human brain the semblance of an interactive network." [p. 139]

The example from China emphasizes that there are gains to be made by being aware of other approaches to pedagogy, as well as to the use of technology.

This is the best chapter in the whole book!

Yellow highlighted passages: Chap. 6 Geniuses and Prodigies

• "[while discussing math prodigies] - can psychology or neurology propose at least an embryo of an explanation for the extraordinary fertility of this unique mind?" [p. 146]
• "The role of memory in mathematics is easily underestimated. ... Their familiarity with numbers is so refined..." [p. 147]
• "The tight link between mathematical and spatial aptitudes has often been empirically demonstrated." [p. 150]
• "Great mathematicians' intuitions about numbers and other mathematical objects do not seem to rely so much on clever symbol manipulations as on a direct perception of significant relations." [p. 151]
• "In the mind of the calculating prodigy, each number does not just light up as a point on a line, but rather as an arithmetical web with links in every direction." [p. 151]

Overall this chapter did not have much to offer, at least on the first reading.

Yellow highlighted passages: Chap. 7 Losing Number Sense

• "Cerebral lesions of various origins can have a devastating and sometimes surprisingly specific impact on arithmetic abilities." [p. 175]
• "The extreme modularity of the human brain stands out as the main lesson to be gathered from studies of cerebral pathology." [p. 177]
• "For most researchers, these models are but provisional metaphors." [p. 200]
• "Prefrontal areas comprise a multitude of networks." [p. 200]

Mildly interesting.

Yellow highlighted passages: Chap. 8 The Computing Brain

• "... many recent experiments in brain imaging are conceived in a 'neo-phrenological' framework. Their only objective seems to be the labeling of cerebral areas." [p. 216]
• "We have every reason to think that the brain does not work this way. Even seemingly simple functions call for the coordination of a large number of cerebral areas. ... Ten or twenty cerebral areas are activated when the subject reads words, ponders over their meaning, imagines a scene, or performs a calculation. ... Only by combining the capacities of several million neurons, spread out in distributed cortical and subcortical networks, does the brain attain it s impressive computational power." [p. 217]
• "If possible, we would want to obtain a new image of brain activity every hundredth of a second. ... PET scanning ... its excellent spatial resolution is accompanied by a deplorable temporal resolution. Each image depicts the average blood flow over a period of at least forty seconds." [p. 221]

The comment on PET scans certainly undermines their value in cognitive research.

Yellow highlighted passages: Chap. 9 What is a Number?

• "A mathematician is a machine for turning coffee into theorems". [p. 231]
• "The preceding chapters abound in counterexamples that suggest that the human brain does not calculate like a 'logical machine'" [p. 233]
• "The inadequacy of the brain-computer metaphor is almost comical. In domains in which the computer excels - the faultless execution of a long series of logical steps - our brain turns out to be slow and fallible. Conversely, in domains in which computer science meets its most serious challenges - shape recognition and attribution of meaning - our brain shines by its extraordinary speed." [p. 234]
• "In a recent book called Descartes' Error, the neuropsychologist Antonio Damasio demonstrates how emotions and reason are tightly linked." [p. 235]
• "If educational psychologists had paid enough attention to the primacy of intuition over formal axioms in the human mind, a breakdown without precedent in the history of mathematics might have been avoided [modern math curriculum]." [p. 241]
• "The child's brain, far from being a sponge, is a structured organ that acquires facts only insofar as they can be integrated into its previous knowledge. It is well adapted to the representation of continuous quantities and to their mental manipulation in an analogical form." [p. 241]
• "Thus, bombarding the juvenile brain with abstract axioms is probably useless. A more reasonable strategy for teaching mathematics would appear to go through a progressive enrichment of children's intuitions, leaning heavily on their precocious understanding of quantitative manipulations and of counting. ... Then, little by little, one may introduce them to the power of symbolic mathematical notation ... but at this stage, great care should be taken never to divorce such symbolic knowledge from the child's quantitative intuitions." [p. 241]
• "Numbers, like other mathematical objects, are mental constructions whose roots are to be found in the adaptation of the human mind to the regularities of the universe." [p. 252]

Descartes Error is another of my favorite books!

Psychology Index