Multiple Mathematical Intelligences
Machine Traced to 2 Circuits.'' This was the New York Times' take
Blakeslee, May 11) on a report in the May 7 Science
by S. Dehaene, E. Spelke, P. Pinel, R. Stanescu and S.Tsivkin.
The report, ``Sources of Mathematical Thinking: Behavioral and
Brain-Imaging Evidence,'' demonstrates that there are at least
two different loci in the brain involved in arithmetic, and that the
two loci do different things.
- Exact calculation is
accompanied by a ``large and strictly left-lateralized
activation in the left inferior frontal lobe.''
- Approximate calculation tasks activate a region in the
bilateral inferior parietal lobule.
The mathematical mind: exact calculation locus in the
left inferior frontal lobe, approximate calculation
locus in the bilateral inferior parietal lobule.
These two loci are also associated with other mental activities:
The report concludes with a discussion of the evolutionary
difference between exact calculation and approximate.
The left inferior frontal lobe is also
activated during verbal association tasks; the behavioral
aspect of this research, a set of experiments with bilingual
subjects, shows that in fact exact calculation
is language-linked, whereas approximate calculation is not.
The authors remark that bilateral inferior parietal loci
are also activated during visually guided hand and eye movements
and mental rotation. In an essay in the same issue of Science,
Brian Butterworth points out that the proximity of the areas
that control finger movements and approximate calculations
``raises the possibility that these brain regions contribute to
finger counting and finger calculation ... and prompts the suspicion
that the parietal lobes, in the course of development and learning,
come to support the digital representation of numbers.'' Presumably
this refers to a pre-verbal ideation of number, with the first three
fingers coming to represent the equivalence class 3, etc.
Dehaene goes deeper into the philosophical, humanistic and
pedagogical implications of
his research in an Edge piece:
What Are Numbers, Really? A Cerebral Basis For Number Sense.
``Symbolic arithmetic is a cultural invention specific to
humans, and its development depended on the progressive improvement of
number notation systems.'' i.e. of mathematical language.
``Approximate arithmetic, in contrast, shows no dependence
on language ... An interesting, though clearly speculative,
possibility is that this language-independent representation
of numerical quantity is related to the preverbal numerical
abilities that have been independently established in various
animal species and in human infants.''
- ``Together, these results may
indicate that the human sense of numerical quantities has a long
evolutionary history, a distinct developmental trajectory, and
a dedicated cerebral substrate. In educated humans, it could
provide the foundation for an integration with language-based
representations of numbers. Much of advanced mathematics may
build on this integration.''
Multiple Intelligences. From the point of view of teaching and
learning mathematics, it is interesting to
situate this research in the context of ``multiple intelligences.''
This is a
circle of ideas developed by Howard Gardner in a series of
books starting with Frames of Mind : The Theory of Multiple
Intelligences in 1983. (Gardner also is featured in an
Truth, Beauty, and Goodness: Education for All Human Beings:
A Talk With Howard Gardner.) His classification is based
on an empirical effort to understand all human mental abilities.
The seven intelligences he proposed:
are each ``a capacity, with its component
processes, that is geared to a specific content in the world
(such as musical sounds or spatial patterns)''. This quote
is from his
Reflections on Multiple Intelligences: Myths and Messages, where
he also mentions that ten years later he proposes adding to the list
an eighth intelligence, that
The experimental results
of Deheane, Spelke and their collaborators strongly suggest that
mathematical ability must involve at least
three of Gardner's intelligences: the logical-mathematical
(by definition) the spatial and the linguistic.
The concept of multiple mathematical intelligences was put
forth, almost at the same time, in a work called Math for Smarty Pants
by Marilyn Burns (Little, Brown & Co., 1982). She remarks: ``being smart in
math can mean several things, and different things.''
When I came across Burns' book I was very pleased to see
explicitly written out, for the first time,
what I had felt as a mathematician for many years: there are
different kinds of mathematical minds. Besides helping me make
sense of a career spent in departments with logicians, analysts
and algebraists it made me very receptive to the ``rule of three''
as propounded by the members of the Harvard Calculus Consortium,
which has become the Rule of Four in the second edition of their
Calculus: ``Where appropriate, topics should be presented
geometrically, numerically, analytically and verbally.'' It
makes sense to access as many intelligences as are available.
- ``There are some kids who are whizzes at dealing with
numbers. They do arithmetic fast, really fast. ...''
- ``Then there are the kids who are great with shapes, who
can `see' things easily in their heads. ...''
- ``Some kids are good at strategy games and puzzles that don't
have much to do with either arithmetic or shape, but have more
to do with thinking logically to figure things out. ...''
and Multiple Mathematical Personalities?
One aspect of the phenomenon that neither Gardner nor the
Dehaene-Spelke team seem to have gone into is the correlation
between the various intelligences and personality traits. In
a department of research mathematicians where the mathematical
intelligences, at least, can be assumed to be present in their
most extreme form, there is considerable evidence of a
correlation, which may not be noticeable in the general
population, with various personality types. Without going into
too many specifics, let me tell the story of a mathematician, a
male, who changed fields early in his career. When his wife was
asked at a cocktail party what her husband did (this probably
dates the story) she said: ``He's a logician.'' ``But,'' she
quickly added, ``he used to be a topologist.''
© copyright 1999, American Mathematical Society.