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. 3 The Adult Number Line

• "Dozens of human societies around the world have progressivley converged on the same solution. Nearly all of them have agreed to denote the first three or four numbers by an identical number of marks, and the following numbers by essentially arbitrary symbols. Such a remarkable cross-cultural convergence calls for a general explanation." [p. 65]
• "It takes less than half a second to perceive the presence of one, two, or three objects. Beyond this limit, speed and accuracy fall dramatically." [p. 67]
• "The numbers 1, 2, and 3 seem to be recognized without any appearance of counting. ... the subitizing ability ..." [p. 68]
• "When viewing concentric circles, for instance, we have to count in order to determine whether there are two, three, or four of them. Thus, the subitizing procedure seems to require objects to occupy distinct locations. ... I therefore believe that subitizing in human adults, like numerosity discrimination in babies and animals, depends on circuits of our visual system that are dedicated to localizing and tracking objects in space." [p. 68]
• "... we all tend to overestimate numerosity when the objects are regularly spread out on a page, and conversely we tend to underestimate sets of irregularly distributed objects." [p. 71]
• "...our perception of large numbers follows laws that are strictly identical to those that govern animal numerical behavior. We are subject to a distance effect [80 and 100 dots are easier to distinquish than 80 and 82 dots] and a magnitude effect [ 90 and 100 dots are more difficult to distinquish than 10 and 20]" [p. 71]
• "Obviously, what distinquishes us from other animals is our ability to use arbitrary symbols for numbers, such as words or Arabic digits. These symbols consist of discrete elements that can be manipulated in a purely formal way, without any fuzziness." [p. 73]
• "... each time we are confronted with an Arabic numeral, our brain cannot but treat it as a analogical quantity and represent it mentally with decreasing precision ..." [p. 73]
• "An Arabic numeral first appears to us as a distribution of photons on the retina, a pattern identified by visual areas of the brain ... the brain hardly pauses at recognizing digit shapes. It rapidly reconstructs a continuous and compressed representation of the associated quantity. This conversion into a quantity occurs unconsciously, automatically, and at great speed. ... Understanding numbers, then, occurs as a reflex." [p. 78]
• "The finding of an automatic association between numbers and space leads to a simple yet remarkably powerful metaphor for the mental representation of numerical quantities: that of the number line." [p. 81]
• "The direction of association between numbers and space seems to be related to the direction of writing. ... the organization of our Western writing system has pervasive consequences on our everyday use of numbers." [p. 82]
• "If we did not already possess some internal non-verbal representation of the quantity 'eight', we would probably be unable to attribute a meaning to the digit 8. We would then be reduced to purely formal manipulations of digital symbols in exactly the same way that a computer follows an algorithm without ever understanding its meaning." [p. 87]
• "The number line ... encodes only positive integers ... perhaps this is the reason for our lack of intuition concerning other types of numbers ... these mathematical entities are so difficult for us to accept and so defy intuition because they do not correspond to any preexisting category in our brain." [p. 87]
• "To function in an intuitive mode, our brain needs images - and as far as number theory is concerned, evolution has endowed us with an intuitive picture only of positive integers." [p. 88]

I'm not sure I agree with his statements about the number line only containing positive integers. My image of it contains all real numbers and even transcendental numbers. By adding a second dimension I also have complex numbers, and in principle I can extend this to n dimensions.

Yellow highlighted passages: Chap. 4 The Language of Numbers

• This is the first of three chapters that compose a major section called 'Beyond Approximation'.
• "How did Homo sapiens alone ever move beyond approximation? The uniquely human ability to devise symbolic numeration systems was probably the most crucial factor. Certain structures of the human brain that are still far from understood enable us to use any arbitrary symbol, be it a spoken word, a gesture, or a shape on a paper, as a vehicle for a mental representation. Linguistic symbols parse the world into discrete categories." [p. 91]
• "When our species first began to speak, it may have been able to name only the numbers 1, 2, and perhaps 3. Oneness, twoness, and threeness are perceptual qualities that our brain computes effortlessly, without counting." [p. 92]
• "How did human languages ever move beyond the limit of 3? The transition toward more advanced numeration systems seems to have involved the counting of body parts. ... Historically then, digits and other parts of the body supported a body-based language of numbers, which is still in use in some isolated communities." [p. 93]
• "It is highly impractical to learn an arbitrary name for each number. The solution is to create a syntax that allows larger numerals to be expressed by combining several smaller ones. ... the choice of a base system" [p. 94-95]
• "Beyond giving numbers a name, to keep a durable record of them was also vital. ... humans quickly developed writing systems that could maintain a permanent record of important events, dates, quantities or exchanges. [p. 95]
• "The principle of one-to-one correspondence has been reinvented over and over ... By adding together several of these symbols, other numbers may be formed. ... This additive principle, according to which the value of a number is equal to the sum of its component digits, underlies many number notations." [p. 97]
• "In retrospect, it seems obvious that addition alone cannot suffice to express very large numbers. Multiplication becomes indispensable. One of the first hybrid notations, mixing addition and multiplication, appeared in Mesopotamia over four millennia ago." [p. 97]
• "One final invention greatly expanded the efficacy of number notations: the place-value principle. ... Place-value coding is a must if one wants to perform calculations using simple algorithms." [p. 98]
• "The peculiar disposition of our perceptual apparatus for quickly retrieving the meaning of an arbitrary shape, which I have dubbed the 'comprehension reflex', is admirably exploited in the Indian-Arabic place-value notation. This numeration tool, with its ten easily discernible digits, tightly fits the human visual and cognitive system." [p. 100]
• "Or are some number notations better adapted to the structure of our brains? Do certain countries, by virtue of their numeration system, start out with an advantage in mathematics?" [p. 102]
• "Rigorous psychological experiments ... repeatedly demonstrate the inferiority of English or French over Asian languages." [p. 102]
• "Chinese number words tend to be shorter ... Our memory span is thus determined by how many number words we can repeat in less than two seconds." [p. 102]
• "The organization of spoken Chinese numerals directly parallels the structure of written Arabic numerals. Hence Chinese children experience much less difficulty than their American counterparts in learning the principles of place-value notation in base ten. ... Base ten is a transparent concept in Asian languages, but it is a real headache for Western children." [p. 105]
• "Western numeration systems are inferior to Asian languages in many respects - they are harder to keep in short-term memory, slow down calculation, and make the acquisition of counting and of base ten more difficult." [p. 105]
• "If children could vote, they would probably favor a widespread reform of numerical notations and the adoption of the Chinese model." [p. 106]
• "At least a prime number such as 7 or 11, or perhaps a number with many divisors such as 12, should have been selected as the base of numeration." [p. 117]

The forming of major sections, the first being about how we are hardwired for approximate analog numerosity and now the second which is an introduction to the way our symbol system allows us to become precise is an important distinction.

Teachers who have no understanding of human learning are analogous to students who have no understanding of arithmetic but proceed to work within the domain by rote algorithms.

Triangles and tetrahedrons ( 3 points or 4). Four is 3-dimensional, which is far richer a metaphor. Now for the nodes: mathematics, psychology, education, and technology. Perfect! Stable and a firm foundation for both learning and teaching. Even students should be made aware of such a metaphor.

The last point is a superb jumping off point for mathematical exploration! Why pick a prime, or its opposite, a number with many factors? What are the properties of each?

Psychology Index