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Imagery The Sensory-Cognitive Connection for Math
Nanci Bell and Kimberly TuleY
Why can’t everyone think with numbers? Why do some children learn math
readily, handle money and time concepts with ease, retain information
from year to year, and think with numbers effortlessly? What cognitive
processes do some have that others do not?
Mathematics is cognitive process-thinking-that requires the dual coding
of imagery and language. Imagery is fundamental to the process of
thinking with numbers. Albert Einstein, whose theories of relativity
helped explain our universe, used imagery as the base for his mental
processing and problem solving. Perhaps he summarized the importance of
imagery best when he said, “If I can’t picture it, I can’t understand
For the people who “get” math, the language of numbers turns into
imagery. They use internal language and imagery that lets them
calculate and verify mathematics; they “see” its logic.
Imaging is the basis for thinking with numbers and conceptualizing
their functions and their logic. The Greek philosopher Plato said, “And
do you not know also that although they [mathematicians] make use of
the visible forms and reason about them, they are thinking not of
these, but of the ideals which they resemble…they are really seeking to
behold the things themselves, which can be seen only with the eye of
The relationship of imagery to the ability to think is one of the
preeminent theories of human cognition. Allan Paivio, author of the
Dual Coding Theory (DCT) and a cognitive psychologist, stated,
“Cognition is proportional to the extent that mental representations
(imagery) and language are integrated.” Research from the 1970s and
into the 1990s has validated Dr. Paivio’s work as a viable model of
human cognition and its practical, as well as theoretical, application
to the comprehension of language (Bell, 1991). Dr. Paivio believes that
in order to think and understand, humans must be able to simultaneously
generate imagery and corresponding language to describe that imagery.
Mathematics is the essence of cognition. It is thinking (dual coding)
with numbers, imagery and language; reading/spelling is thinking with
letters, imagery and language. Both processes, often mirror images of
each other, require the integration of language and imagery to
understand the fundamentals and then apply them. Dual coding in math,
just as in reading, requires two aspects of imagery: symbol/numeral
imagery (parts/details) and concept imagery (whole/gestalt).
Visualizing numerals is one of the basic cognitive processes necessary
for understanding math. For example, we image the numeral “2” for the
concept of two. When we see the numeral “3,” we know that it represents
the concept of three of something: three pennies, three apples, three
horses, three dots. If someone gives us two pennies for the numeral
three, we have a discrepancy between our numeral-image for three and
the reality (concept) of three. The first imagery needed for math is
the symbolic (or numeral) imagery that represents the reality of a
What does numeral imagery look like? Here’s one example. Cecil was very
good in math. He could think with numbers, arrive at answers in his
head, and mentally check for mathematical discrepancies in finance or
life situations easily. He explained this ability, “I just visualize
numbers and their relationships. Certain numbers are in certain colors,
and the number-line in my head goes specific directions.” Not only
could Cecil visualize numerals and concepts, both types of imagery, but
he also had an unusual talent for color imagery. He assigned colors to
“What color is the number 14?” he was asked.
His eyes went up, and in all seriousness, he said, “Light
blue.” Similarly, number 3 was reddish pink and the number 88 “kind of
a purple.” Quizzed again months later, Cecil assigned the same colors
to the same numbers. Chronological relationships appear in our minds on
a number line, the days of the week, the months in the year. Imagery is
our sensory systems’ way of making the abstract real. It is a means to
While imaging numerals is important to mathematical computation,
another aspect of imagery is equally important: concept
imagery. Understanding, problem solving and computing in mathematics
require another form of imagery--the ability to process the gestalt
(the whole). Sometimes children or adults can visualize the numerals,
the parts, but cannot bring those parts to a whole, just as they can
sometimes visualize individual words but cannot bring those words to a
whole to form concepts. Mathematical skill requires the ability to get
the gestalt, see the big picture, in order to understand the process
underlying mathematical logic.
“Concept imagery is the ability to image the gestalt (whole),” Bell
(1991). Concept imagery is basic to the process involved in oral and
written language comprehension, language expression, critical reasoning
and math. It is the sensory information that connects us to language
The ability to create mental representations for mathematical concepts
is directly related to success in mathematical reasoning and
computation. However, because some children do not have this imaging
ability, they are often mislabeled as not trying, unable to retain
information, or having dyscalculia (the inability to perform arithmetic
Manipulatives May Not Be Enough
Joanie’s second grade class covered a review of recognizing numbers,
addition, subtraction, and even some multiplication. They worked a lot
with concrete manipulatives and Joanie was doing well at the end of the
year. But her third grade teacher complained that Joanie didn’t know
anything about numbers.
Concrete experiences-manipulatives-have been used for many years in
teaching math (Stern, 1971). However, like Joanie, many children and
adults have often experienced success with manipulatives, but failure
in the world of computation (NCTM, 1989; Moore, 1990; Papert,
1993). They have what has often been described as “application
Joanie’s second grade class had spent a lot of time with
manipulatives. Some of the children moving on to third grade continued
to “think with numbers.” Their experience with manipulatives became
part of their mental deposit of imagery. Like a bank deposit, these
images could be drawn upon at will. However, not all children create
mental imagery as they work with concrete manipulative. For these
children, the process of turning the concrete experience into imagery
must be consciously stimulated.
On Cloud Nine® Math
Concrete to Imagery to Computation
Arnheim (1966) wrote, “Thinking is concerned with the objects and
events of the world we know…When the objects are not physically
present, they are represented indirectly by what we remember and know
about them…Experiences deposit images.”
Numbers can be experienced and the relationships between them can be
made concrete by using manipulatives. What appears abstract can be
experienced and imaged to concreteness. Math’s roots are in the realm
of the concrete, and imagery is the link to mathematical processing,
retention, and application.
To develop concept and numeral imagery, the On Cloud Nine® math program
(developed by the authors) integrates and consciously applies imagery
to the cognitive process of computing and conceptualizing math and
mathematical principles. As individuals become familiar with the
concrete manipulatives, they are questioned and directed to consciously
transfer the experienced to the imaged. They image the concrete and
attach language to their imagery. The integration of imagery and
language is then applied to computation. Individuals develop the
sensory-cognitive processing to understand and use the logic of
The program moves through three basic steps to develop mathematical
reasoning and computation using: 1) manipulatives to experience the
reality of math, 2) imagery and language to concretize that reality in
the sensory system, and 3) computation to apply math to problem
solving. On Cloud Nine® manipulatives serve two purposes: 1) to
concretize numbers and mathematical concepts, and 2) to serve as a base
for establishing imagery.
When asked to add the numbers 3 + 2, children who are drawing on their
vault of images may see 3 apples and 2 more oranges to show 5 pieces of
fruit. Others may draw on an image of a number line and place their
mental finger on the 3 as a starting point. The “+” tells them to move
forward and the “2” indicates how many places. They know the answer
because they can “see it” in their mind’s eye. These children may look
up as they access their images (defocusing).
Children who don’t seem to have a vault of images may say things like
“I don’t remember that one.” They need explicit instruction in imaging
the concrete and applying that imagery to the computation.
How does imaging as a conscious process work? The On Cloud Nine® math
program begins with numbers in isolation—numeral imagery. A student is
asked to view the written numeral, and then it is taken away. The
student must demonstrate the “number” underlying the numeral by showing
how many cubes represent that number. The student sees, says, and
writes the number in the air. The goal is for the student, when she
sees the numeral, to immediately create an image of the formation of
that number and the value behind it.
The process continues with experiencing the number line, first as a
concrete manipulative, then as a flexible mental image. “Show me where
you see the number 15?” “What’s the number one step up from that?” “Is
the 3 close to the 15 or quite far away?” “What number is closer to the
15 – the 10 or the 5?” Students develop a number line they carry with
them in their vault of images. These students can access their vault of
images at will. Conscious imagery and the ability to simultaneously
create images and verbalize these imaging—dual coding—are continued as
children are taught addition, subtraction, word problems,
multiplication, division and more advanced math.
On Cloud Nine® math integrates and consciously applies imagery to the
cognitive process of computing and conceptualizing math and
mathematical principles. Children image the concrete and attach
language to their imagery. The integration of imagery and language is
then applied to every aspect of mathematical computation.
All children can develop the sensory-cognitive processing to understand
and use the logic of mathematics. In every aspect of math, children can
have access to what becomes an innate bank vault of imagery for memory
Nanci Bell, owner and director of Lindamood-Bell Learning
Processes, is the author of two books on imagery as the base for
language processing. Kimberly Tuley, the director of operations
for Lindamood-Bell is a trainer and consultant in the application and
refinement of Lindamood-Bell® programs.
Aristotle. (1972). Aristotle on Memory. Providence, Rhode Island: Brown University Press.
Arnheim, R. (1966). Image and thought. In G. Kepes (Ed.). Sign, Image, Symbol. New York: George Braziller, Inc.
Bell, Nanci. (1991). Visualizing and Verbalizing for Language Comprehension and Thinking. Paso Robles: NBI Publications.
Moore, David S. (1990). On the Shoulders of Giants: New Approaches to
Numeracy. Steen, L. (Ed.). Washington, D.C.: National Academy Press.
Papert, Seymour. (1993). The Children’s Machine: Rethinking School in the Age of the Computer. New York: Basic Books.
Paivio, Allan. (1981). Mental Representations: A Dual Coding Approach. New York: Oxford University Press.
Stern, Catherine and Stern, Margaret B. (1971). Children Discover Arithmetic. New York: Harper & Row, Publishers, Inc.
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