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?
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 it.”
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.
The relationship of imagery to the ability to think is one of the preeminent theories of human cognition. Allan Paivio, renowned cognitive psychologist and author of the Dual Coding Theory (DCT), stated, “Cognition is proportional to the extent that mental representations (imagery) and language are integrated.”
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 number concept.
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 specific numbers! “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 experience math.
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). Mathematical skill requires the ability to get the gestalt, see the big picture, in order to understand the process underlying mathematical logic.
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 operations).
Manipulatives May Not Be Enough
Concrete experiences-manipulatives-have been used for many years in teaching math. However, many children and adults have often experienced success with manipulatives, but failure in the world of computation. They have what has often been described as “application problems.”
Many children have spent a lot of time with manipulatives. As they progress through school they are able 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 manipulatives. For these children, the process of turning the concrete experience into imagery must be consciously stimulated.
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 and computation.