more on the mental number line

in the Wall Street Journal:

Body posture can influence how we estimate such things as age and size, a study shows.

Thirty-three undergraduates stood on a Wii Balance Board, a videogame-system accessory. Researchers surreptitiously manipulated the subjects' stances, slightly tilting them, though an onscreen measure misled the students into thinking they were evenly balanced.

In each stance, students answered 13 questions, including the height of the Eiffel Tower, the size of the Netherlands and the life expectancy of a parrot. On average, participants gave smaller estimates when they leaned left than when they stood straight or leaned right—stances producing virtually identical results.

"Mental-number-line theory" accounted for the finding, the researchers said. We envision numbers as they appear on a ruler, rising from left to right. Leaning left nudged estimates lower.

"Leaning to the Left Makes the Eiffel Tower Seem Smaller: Posture-Modulated Estimation," Anita Eerland, Tulio M. Guadalupe and Rolf A. Zwaan, Psychological Science (December).
Christopher Shea | Week in Ideas

number line posts

The times table and the number line

Stanislas Dehaene on why learning the times tables is hard:

“Number sense” is a short-hand for our ability to quickly understand, approximate, and manipulate numerical quantities.

[snip]

My hypothesis is that number sense qualifies as a biologically determined category of knowledge. I propose that the foundations of arithmetic lie in our ability to mentally represent and manipulate numerosities on a mental “number line”, an analogical representation of number; and that this representation has a long evolutionary history and a specific cerebral substrate.

[snip]

The hypothesis developed in TNS [The Number Sense] is that all children are born with a quantity representation which provides the core meaning of numerical quantity. Exposure to a given language, culture, and mathematical education leads to the acquisition of additional domains of competence such as a lexicon of number words, a set of digits for written notation, procedures for multi-digit calculation, and so on. Not only must these abilities be internalized and routinized; but above all, they need to be coordinated with existing conceptual representations of arithmetic. The constant dialogue, within the child’s own brain, between linguistic, symbolic, and analogical codes for numbers eventually leads, in numerate adults, to an integrated set of circuits that function with an appearance of non-modularity. Before such a flexible integration is achieved, however, the hypothesis of a modularity and lack of coordination of number representations can explain many of the systematic errors or difficulties that children encounter in the acquisition of arithmetic.

[snip]

Here I shall only discuss one example, the memorization of the multiplication table. Why is it so difficult to learn the small number of single-digit multiplication facts? Leaving out multiplications by 0 and by 1, which can be solved by a general rule, and taking into account the commutativity of multiplication, there are only 36 facts such 3x9=27 that need to be learned. Yet behavioral evidence indicates that even adults still make over 10% errors and respond in more that one second to this highly overtrained task.

What I think is happening is that our intuition of quantity is of very little use when trying to learn multiplication. Approximate addition can implemented by juxtaposition of magnitudes on the internal number line, but no such algorithm seems to be readily available for multiplication. The organization of our mental number line may therefore make it difficult, if not impossible, for us to acquire a systematic intuition of quantities that enter in a multiplicative relation. (This hypothesis is supported by the fact that patients can have severe deficits of multiplication while leaving number sense relatively intact; in particular, patient NAU (Dehaene & Cohen, 1991), who could still understand approximate quantities despite aphasia and acalculia, was totally unable to approximate multiplication problems). In order to memorize multiplication facts, we therefore have to resort to other strategies based on nonquantitative number representations. The strategy that cultures throughout the world have converged on is to acquire multiplication facts by rote verbal learning. That is, each multiplication fact is recited and remembered as a rote phrase, a specific sequence of words in the language of teaching. This is still a difficult task, however, because the 36 facts to be learned all involve the same number words in slightly different orders, with misleading rhymes and partial overlap. Error analyses indeed indicate that interference in memory is the most frequent cause of multiplication error. When we err, say, on 7x8, we do not produce 55 or 57, which would be close matches, but we typically say 63, which is the correct multiplication result of the wrong operation, 7x9 (Ashcraft, 1992; Campbell & Oliphant, 1992).

It can be argued that our memory never evolved to acquire a lot of tightly inter-related and overlapping facts, as is typical of the multiplication table. Our long-term semantic memory is associative and content-adressable: when cued with a specific episode, we readily retrieve memories of related contents based on the semantic similarity. In particular, we generalize approximate additions based on numerical proximity. Hence, we can readily reject 34+47=268 as false, even though we have never been exposed to this particular fact, because our representation of quantity immediately allows us to recognize that the proposed quantity, 268, is too distant from the operands of the addition (Ashcraft & Battaglia, 1978; Dehaene et al., 1999). In the case of exact multiplication, however, the organization of memory by proximity is detrimental to performance. It would be desirable to keep each multiplication fact separate from the others ; yet our memory is designed so that, when we think of 6x7, we co-activate 6x8 and 5x7. In summary, our cerebral organization can explain both why exact multiplication facts are so confusing and difficult to learn, and why approximation and understanding of quantities are highly intuitive operations.

Précis of “The number sense”
Stanislas Dehaene
INSERM U.334
Service Hospitalier Frédéric Joliot
CEA/DSV
4 Place du Général Leclerc
91401 cedex Orsay

This is pretty much Wayne Wickelgren's explanation as to why it's difficult to memorize the times tables.

My own systematic error in multiplication, on the other hand, directly contradicts Dehaene's observations: until 5 years ago, I had spent my entire adult life beileiving that 7x6=43. (As I recall, Vlorbik was one of the people who alerted me to the fact that 7x6=42, back on the old site.)

If I were a normal person I would have spent my entire adult live believing that 7x6 was 49 or possibly 35.




The Number Sense: How the Mind Creates Mathematics, Revised and Updated Edition

the number lines: linear vs log

Andrew Gelman:

Among households with dangerous wells (arsenic content higher than 50 (in some units)), we predicted whether a household switches wells, given two predictors:
- distance to the nearest safe well;
- arsenic level of their existing well.
The data were consistent with the model that people weight “distance to nearest safe well” linearly but weight “arsenic level” on the log scale. As we discuss in our book, this makes psychological sense: distance is something you perceive directly and linearly, by walking (it takes twice as much time and effort to walk 200m as to walk 100m), whereas arsenic level is just a number and, as such, going from 50 to 100 seems about the same, psychologically, as going from 100 to 200 or 200 to 400–even though, in reality, that last jump is four times as bad as the first (arsenic being a cumulative poison).

Stanislas Dehaene on the mental number line

My hypothesis is that number sense qualifies as a biologically determined category of knowledge. I propose that the foundations of arithmetic lie in our ability to mentally represent and manipulate numerosities on a mental “ number line ”, an analogical representation of number ; and that this representation has a long evolutionary history and a specific cerebral substrate. “ Number appears as one of the fundamental dimensions according to which our nervous system parses the external world. Just as we cannot avoid seeing objects in color (an attribute entirely made up by circuits in our occipital cortex, including area V4) and at definite locations in space (a representation reconstructed by occipito-parietal neuronal projection pathways), in the same way numerical quantities are imposed on us effortlessly through the specialized circuits of our inferior parietal lobe. The structure of our brain defines the categories according to which we apprehend the world through mathematics. ” (TNS, p. 245).
Précis of “ The number sense ”
Stanislas Dehaene
INSERM U.334
Service Hospitalier Frédéric Joliot
CEA/DSV
4 Place du Général Leclerc
91401 cedex Orsay
France

The Number Sense: How the Mind Creates Mathematics, Revised and Updated Edition

teach the number line in 1st grade

Several times over the past years I've come across the idea that humans possess an innate number line inside our minds. At least, that's how I interpret of the snippets of research I've read.

Not long after encountering the possibility that number lines have a privileged place in math learning, I read H. Wu's revelatory definition of a fraction as a point on the number line:

The following is a new approach to the teaching of fractions. It is not new in the sense of introducing new concepts; the subject is too old for that. Rather, it is new in the way the various skills and concepts are introduced and woven together. Whereas it is traditional to ask you to believe that the concept of a fraction is so profound that you have to be willing to accept multiple meanings for it at the outset, we merely ask you to accept one clear-cut definition of a fraction (as a point on the number line), and use reasoning to deduce as logical consequences all other meanings of this concept.
On the Teaching of Fractions (pdf file)
H. Wu

I've been relying heavily on number lines for self teaching and reteaching for several years now.

David Geary's new longitudinal study seems to add further evidence that number lines are important:

The researchers also found that first-graders who understood the number line and how to place numbers on the line and who knew some basic facts showed faster growth in math skills than their counterparts during the next five years.
MU Psychology Study Finds Key Early Skills for Later Math Learning
Long-term study shows students must know about numbers at beginning of first grade
July 11, 2011