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15 times with their finger while blocking out the
counting system through verbal interference, can
usually perform 15 taps, showing that humans can
approximately map the number 15. As subjects
try to tap higher numbers, responses become
less precise (Carey, 2004). According to Varley,
Klessinger, Romanowski, and Siegal (2005), the
numerical reasoning and language are function-
ally and neuro-anatomically independent in
adult humans. They demonstrated a dissociation
between grammatical and mathematical syntax
in patients with brain damage who suffered from
severe aphasia, preventing them from understand-
ing or producing grammatically correct language.
Patients demonstrated proficiency in mathematical
syntax, despite an inability to comprehend analo-
gous syntax in spoken or written language. The
authors describe the patients who were unable to
differentiate between the statements “Mary hit
John” and “John hit Mary,” but these same patients
successfully solved mathematical operations that
were structurally dependent in this same general
way, for instance, the difference between 52 - 11
and 11 - 52. Similarly, performance of patients
who were unable to comprehend sentences with
embedded clauses was unimpaired in comput-
ing expressions with embedding, for example,
answers to a sequence such as 90 - (3 + 17) ×
3. Language disorders and calculation abilities
occur independently of each other (Gelman &
Butterworth, 2005), which have been shown
with the use of neuroimaging - techniques for
picturing the structures of functions in the brain.
Rhesus monkeys can represent the numerals one
through nine on an ordinal scale (Brannon and
Terrace, 1998). Models of number processing,
which are language-independent, indicate that
number concepts exist prior to verbal counting
(Bar-David et al., 2009).
Several research studies investigated the senses
and brain areas involved in numeral activities in
children, first without naming, then with names
spoken, read, or seen as written words. Young
children have a limited understanding of a concept
of number. Children start out as “subset-knowers:”
1-knowers (i.e., they can give one exact object
when asked for one), then 2-knowers, 3-knowers,
and 4-knowers (Wynn, 1992). Children that can
give a requested number of objects (for example,
yellow rubber ducks from a green bowl) within
their productive count range are labeled as cardinal
principle (CP)-knowers. They fall into two groups:
mappers and non-mappers (LeCorre & Carey,
2006). Mappers are able to metaphorically map
an accurate verbal number estimate to an observed
array of objects while non-mappers can only do this
for quantities under four. Mappers have an intuitive
sense of quantity for each mapped number word
and thus an understanding of the logic of the count
list. The subset-knowers do not truly understand
the concepts that the number words represent
outside of a certain range, while CP-knowers do.
Bar-David et al. (2009) explored the relationship
between number language and the representation
of exact large numbers. They analyzed the cogni-
tive tools that enable children develop number
concepts. They tested both subset- and CP-knowers
on a nonverbal task involving numbers, both
small (less than four) and large (greater than four)
numbers. Among many other tests, they authors
asked children to retrieve “just enough socks”
for caterpillars with varying quantities of feet.
The authors propose two interpretations of their
results: (1) the accumulation of verbal knowledge
by a subset-knower leads to the development of
more precise nonverbal number structures which
scaffolds and enables the induction to CP-status,
or (2) the accumulation of verbal knowledge by
a subset-knower enables the induction directly,
and this induction facilitates the development of
non-verbal number concepts.
According to Halberda, Mazzocco, & Feigen-
son (2008), adults, infants, and animals share a
sense of a quantity, while competence in domains
such as calculus emerges from a different system,
as it relies on symbolic representations that are
unique to humans who received instruction. An
ancient evolutionary number system supports basic
