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