Improve Grades

Posts Tagged ‘Improvement’

At Arlington, they ‘Believe’ improvement will occur LAWRENCE — Sitting on a rug, fourth graders at Arlington Elementary School listened intently as Brigid Hegarty read “A Chair for my Mother.” After telling students how the mother, daughter and granddaughter in the story were saving their money, she stopped and asked the children what they would save their money for. Read more on The Eagle-Tribune

THE CLIPBOARD: Continuous school improvement in 2011 Keyser, W.Va. — Well, it’s another New Year, a time we reflect on the past year and make resolutions for the new year. I feel I met last year’s professional goal of resuming the writing of The Clipboard for the Mineral Daily News-Tribune. Read more on Mineral Daily News-Tribune

Information, support and audio/digital resources for women seeking happiness and success in every area of their lives. Generous 50% commissions on recurring memberships. Affiliate center filled with useful tools/information for promoting our products. Self Improvement & Transformational Tools For Women-50% Recurring

Bill Austin has placed some of his most powerful products for healing, personal growth and abundance on the CB network! Digital Products for Healing, Self Improvement and Success

MIDDLETOWN SOCCER PREVIEW: Girls, boys teams seek improvement in 2010 MIDDLETOWN — The two soccer programs at Middletown High are far apart. The girls soccer team returns a ton of players and its goal is not only to qualify for the state tournament for the third year in a row, but to go deep into the postseason event. Read more on The Middletown Press

This handbook focuses on the use of feedback and evaluation forms to develop information about your teaching skills and style. Please recognize that evaluation and feedback forms are only part of an overall continuous teaching improvement process. Other methods for obtaining useful feedback to improve your teaching include, but are not limited to: soliciting verbal feedback from your students, having your lecture videotaped, or arranging peer or supervisor class observation. These and other methods are briefly summarized in the Handbook, but the focus will be on the evaluation forms and resources to help you improve.

For those of you familiar with the Plan, Do, Check, Act cycle and other quality improvement principles, you will notice the framework described above reflects this same cycle:

PLAN: As a teaching assistant, you prepare classroom instruction.

DO: You teach the class or lead the discussion or lab.

CHECK: You evaluate your own performance. The evaluation process outlined in this Handbook can be used to accomplish this step. During the “Check” phase, you’ll discover things you do effectively, as well as opportunities for improvement.

ACT: To complete the cycle, you need to determine how to use the feedback and what steps you need to take to improve the teaching and learning associated with your class. The Resource Guide suggests books, articles, videotapes of past workshops, and future workshops.

At this point the process starts over. The concept is continuous improvement. As TA(Teaching Assistant) Fellows, we encourage you and all faculty to adopt a continuous improvement philosophy toward teaching. Note that once you have asked the students: “How can I improve?” they will expect you to improve, just as you expect them to learn.

In the spirit of continuous improvement and the environment of trust, accountability, and goodwill, the College of Engineering TA Fellows believe the following framework of feedback forms and corresponding resources will help TAs continuously improve undergraduate education in the College.

Evaluation and Improvement Process Overview

Fast Feedback
Several types of fast feedback exist, ranging from informal conversation with students to the use of fast feedback forms. Some forms are designed to be used on the first day of class, others are designed to be used after a few weeks of class, or at various times throughout the semester. The use of fast feedback forms is quite flexible. With fast feedback, you can familiarize yourself with your class and their expectations, and you can identify and correct simple problems in areas such as style, presentation, or facilities.

Mid-Semester Evaluations
Mid-semester is an ideal time to obtain feedback from your class to help you improve. At this point in the semester, students in your class or lab have been exposed to homework, exams, labs, projects, and your grading, and may have useful feedback on your teaching in these contexts. The objective is to attempt a midsemester correction that leads to high final quality performance indicated by the end-of-semester evaluations.

Classroom observation and TA evaluation by course supervisors is now required by the College of Engineering for all new TAs (first and second semester) by the seventh week. A sample mid-semester evaluation form is provided in this Handbook that you can provide to your course supervisor for this purpose. Even if you are not a new TA or if your department does not require it, you may wish to implement this step. Since all TAs are required by COE to be evaluated by their supervisor at the end of the semester, it isimportant that all TA’s be observed in a teaching situation at some point during the semester in order for thesupervisor to assess teaching performance appropriately.

End-of-Semester Evaluations

The end-of-the semester evaluation is generally not as flexible as the previous two steps. Most departments require the use of a particular, standardized end-of-semester student evaluation form. The purpose of this evaluation is two-fold: (1) to obtain an overview of your strengths and weaknesses throughout the semester so you may improve next semester, and (2) to provide a formal performance assessment so your department can both ensure high teaching standards, and in some cases, determine whether to reappoint you.

A standardization of the forms is necessary for fair TA assessment. The TA Fellows have spent considerable time developing an end-of-semester form specifically for TAs that combines the best aspects of the existing forms. Teaching evaluations have been separated from course evaluations, and distinction is made between laboratory and lecture/discussion environments. The Academic Planning Council will review the TA Fellows’ recommended forms in the Fall of 1994.

Advice for effective use of evaluations

· Give the evaluation forms out at the beginning of class
Giving the forms at the end of the lecture is an invitation for the students to rush through the form so they can leave sooner. Reserving time at the beginning of a class for evaluations shows that you are serious about receiving feedback and can lead to increased participation.
· Clearly state the purpose of the form
Honestly explaining exactly what the evaluation form will be used for and who sees it can lead students to provide the most useful information. For example, it might be useful to explain when a form will be seen only by the TA versus when the form will be analyzed by faculty or the department administration.
· Read the instructions aloud to your class
Quality of the feedback is ensured only if students fully understand the form. For example, you should clearly explain the “grading” scale on any form that has one. A score of “1″ may mean “excellent” on one type of form while it may mean “needs much improvement” on another. While the forms in this Handbook should be consistent in this respect, you must remember that students fill out forms for different departments and colleges all over campus.
· Review important comments from previous forms with the class
If a noteworthy point or issue is raised on a particular in-semester form, it may be helpful to specifically raise the issue during a subsequent class. This will show the students that you read the forms and that you are prepared to act on the information in them, or at least comment on the particular issue.
· Develop alternative questions and comments
You are strongly encouraged to develop your own evaluation forms to suit your needs and teaching style. Appendix II to this Handbook contains a list of alternative questions and comments that you might find useful in developing your own types of feedback or evaluation.

Effective college study skills are critical for college success. There is no one size fits all method of studying, but one thing that is likely true: the study habits you had in high school will probably be inadequate at the college level. Developing good study skills from the beginning will start you off on the path toward success in your college years.

Study the material before you go to class. This may not have been necessary in high school. But with the amount and depth of material covered in each class at the college level, you will be behind before class even starts if you do not take time to become acquainted with the material before you step into the classroom. You will be relieved, if not pleasantly surprised, at how much easier you will absorb the lecture in class when you have made the effort to familiarize yourself with the subject beforehand.

Study every day. Studies show that students who take time to study every day fare better than students with more sporadic study habits, and much better than those who cram. How much time should you study? A rule of thumb for effective college study skills is to devote three hours a week for homework and studying for each credit hour. If your biology class is three credit hours, then you should spend approximately 15 hours a week on homework and studying outside the classroom. If you are taking a total of 15 credit hours, then you should be spending upwards of 45 hours per week on homework and studying. When you think about it, committing to a daily study schedule is really the only option you have for getting in all the hours needed for studying and homework each week.

Take an active role with your studying. College study skills require more than just reading. Highlight the material, take notes, make outlines, even quiz yourself or have a friend quiz you. Take a cue from grade school and use flashcards. These are good strategies to apply whether you are studying new material to prepare for class, studying after class for reinforcement, or preparing for an exam. Actively studying on a daily basis will help you comprehend and retain the material.

Adjust your study habits as needed. In the beginning you may go through trial and error as you figure out a system that works. You may find that some classes do not require as much studying as you originally thought but others require more of your time than you anticipated. Keep making the necessary changes until you have developed a system that meets your needs.

Plan, plan, plan. Effective time management is your best ally. Good college study skills require managing your time, and this takes planning. Take the time to plan and organize your days and weeks to allocate sufficient time for homework, studying, and other critical tasks. This is essential. If you are taking a full course load, then the amount of time you must devote to homework and studying each week is the equivalent of a full time job, and that is in addition to the time you spend in class. If you do not have a schedule to manage all of this then you will suffer for it.

Plan the first week and see how things go. Make any necessary adjustments for the following week, and continue adjusting until you have devised a plan that works. Include 15 minutes before the end of each day to review the next day’s schedule. You will be surprised at how helpful a step this can be. Doing this before the end of the day will come in handy in case you overlooked something you need for tomorrow that must be taken care of today. At the very least you will minimize surprises, even save yourself some grief, and at best, you will be totally prepared for tomorrow’s undertakings. A word of caution: do not schedule your days so tightly that you have no flexibility to deal with the unexpected. Leave a little wiggle room to accommodate those unanticipated situations that inevitably crop up.

With some up-front investment in time and effort, you can acquire effective college study skills that will pay off with good grades.

Are you looking to decrease your swimming times, and win more swim races. Strength alone will not do the trick. You need good technique and the best way to ensure good technique is to use the power of your mind through hypnosis – 75% for each sale! Swimming Improvement Products

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/

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/