Math Manipulatives Teaching
Math Manipulatives Teaching
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As Published:
Ld-Online.org
Washington Parent Magazine
Imagery The Sensory-Cognitive Connection for Math
Nanci Bell and Kimberly TuleY
http://www.lindamoodbell.com/
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 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.
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 mind?”
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).
Numeral Imagery
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.
Concept Imagery
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 and thought.
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
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 problems.”
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 mathematics.
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 and computation.
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.
Bibliography
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.
More Information:
http://www.lindamoodbell.com/
http://inforequest.lblp.com/
About the Author
http://www.lindamoodbell.com/
http://inforequest.lblp.com/
Pica...does anyone know anything about it?
I'm a teacher and I have a student who eats very strange things: he will eat dirt, tissues, cotton, sticky tack, etc. He has even eaten little tiny cubes that we use as manipulatives in math class. The strange thing is, he eats bizarre things and then isn't hungry for real, normal, good food at lunch time. We had a pizza party in our classroom with pizza, chips, pop, etc (food that most kids love!). He wasn't interested at all. He ate 2 bites and said he was full. I'm just wondering if these are some symptoms of Pica? On a side note, I teach special ed. and he has Aspergers Disorder...is Pica sometimes related to Autism or Aspergers? I'm worried about him, and would like to know more about the disease. Thank you
I'm not an expert but I studied Abnormal Psychology both here and in Australia. Pica is my second most interesting symptom of pscyhological phenomenon. (Ticks associated with Tourettes is my first, but that's another question.)
Pica can be a symptom of another issue, or it can be just a stage of experimentation. Rarely is it a lifetime affliction. About 25% of children with symptoms of developmental obstacles exhibit pica lasting at least a few months. It's really common in people who have obsessive compulsive disorders so I wouldn't be surprised if it was highly correlated with Aspergers.
If this is something the student started doing recently it may be that he's experimenting, feeling with his mouth instead of hands, which may go away after he's satisfied his discovery. Not technically pica, but still could be something to be concerned about until his fixation switches.
If the pica is more persistant than that, it may be something that needs behavioral coaching and much more alert supervision (as if you aren't already an eagle-eye with the class, right?) Obviously you're right to be concerned about what he's eating and clearing his area of stuff that he shouldn't be eating is a good idea. Longer term treatment can vary depending on the person. It may be treated by behavior modification (like you would teach a student not to hit, you teach them not to eat nonedible things). It may just need to be added to healthy eating curriculum, as a matter of education.
You'll definitely want to mention it to a caregiver or their doctor if you have contact with them. Pica is sometimes a sign that the person is not getting enough of essential nutrients like iron, calcium, etc. If not they'll still want to check to make sure he hasn't ingested anything that would cause a intestinal blockage or perforation. You and they will want to make sure the students environment is clear of chemicals and toxic or dangerous susbstances (including sharp objects, magnets, food wastes, etc.)
The good news is most pica is temporary and most children will grow out of it, most people with developmental impairments can be coached out of it eventually.
Hope this helped. Good luck!
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