MAKING MATH REAL®:
A Multisensory Structured Program For Integrating
Sensory-Cognitive Development with Conceptual/Procedural Instruction
By David Berg, E.T.
Founder/Director of the Making Math Real Institute
Creator of the Making Math Real Multisensory Structured Methodologies in Mathematics, K-12
Students with math anxiety do not lack the intelligence or the motivation to be successful. Typically, they lack the underlying development that supports the acquisition of the basic tools to do math. For example, a significant number of students who struggle with math have a “big picture”learning style that enables them to do well with reading comprehension, the conceptual side of math, and other learning activities that create pictures in the mind. These students are gifted image makers. They have vivid imaginations and are often highly creative and expressive. However, they may lack the ability to be successful with the mechanics/basic skills of math such as learning the math facts and being accurate with their calculations. They suffer great anxiety because they know their performance does not match their intellectual capability. Students’ anxiety is exacerbated further by current curricula or programs designed to be implemented at a rate that is too fast for them to process effectively. While they are still struggling to synthesize yesterday’s lesson, somehow they have to contend with today’s new content. Consequently, these children are unable to build a strong foundation in math, but do increase their anxiety and negative association to the subject. They know they are just as smart as their classmates, why can’t they do as well? One reason is big picture learners are not as adept at processing symbols (numbers) as they are at processing pictures.
Some of the key underlying developmental tools essential for processing symbols, learning math facts, and gaining accuracy with calculations are symbol imaging, detail analysis and sequential processing. Symbol imaging is a perceptual tool for imaging and holding sequences of symbols (numbers)in the mind allowing students to perceive and then store information in active working memory for immediate application. In addition, this perceptual tool allows students to input and output information from long term memory. Detail analysis requires the integration of symbol imaging to enable students to focus on the discreet parts of a whole without losing the picture of the whole. Detail analysis is a cognitive editing tool. It supports the ability to notice “careless” errors and check one’s work. Sequential processing, which further integrates symbol imaging and detail analysis, is the ability to recall, re-image, and reconstruct procedures in their respective sequences for accurate application. The focus of the Making Math Real approach is to integrate these three principal sensory-cognitive abilities into every lesson.
Making Math Real is a simplified and practical model that is designed to reach the full diversity of learning styles. It is a systematic, incremental, multi-sensory methodology that guides students from the concrete to the semi-concrete to the semi-abstract, culminating in the synthesis of abstract functioning. While it is successful for the special needs population, its application in general education classrooms provides accelerated, in-depth learning that addresses the educational needs of all students. The focus is in reconnecting math to its concrete fundamentals while developing essential memory building tools to make math an anxiety-free, successful, and dynamic experience for teachers and students.
Making Math Real eliminates math anxiety through authentic experiences of success by reducing reliance on memory, providing multisensory incremental guidance and scaffolding of all math content, and developing the brain tools necessary for success with basic skills. This multisensory structured program reduces reliance on memory by connecting math terminology (symbolic, non-pictorial) to informal language that creates clear mental pictures. Students learn imagistic stories that recreate the concrete experience of math rather than memorizing the steps to an operation. They find success because they see and understand what they are really doing rather than following a rote procedure.
A brief example from Level 2, within the eight levels of the long division sequence, demonstrates how to link informal language to a concrete experience. Long division starts with a concrete and imagistic story of kids (the divisor) finding an abandoned box of some desirable loot (the dividend) such as video games. The video games are packaged in ten packs and single packs. Each kid wants to know how much s/he is going to get (the quotient). Base ten manipulatives are used to represent the loot, tens rods represent ten packs of video games and unit cubes represent single video games. Other manipulatives that represent the kids (the divisor) are placed outside the loot box. The long division algorithm is primarily about place value, so the loot pile (dividend) is organized into separate piles of like value: ten packs together in one pile and singles together in a separate pile. When dividing up valuable loot, most people would be interested in starting with the most valuable pile (the largest place value pile), so the kids start dividing up the ten packs first each saying “one for me, two for me” as they go. Students record the results of each division of the loot using color coding to differentiate and link each place value of the dividend with its respective place value in the quotient. The algorithm of long division (divide, multiply, subtract, check and bring down) recreates this concrete experience. 1. DIVIDE: Each pile is divided up equally to see how much loot each kid gets. 2. MULTIPLY: Multiply shows how much all the kids get by combining how many times each kid gets a video game from the loot pile. 3. SUBTRACT: This step shows the kids taking away all of their loot from the loot originally in the box. 4. CHECK: This is the most important step to make sure that the kids have not been cheated. The left over video games after subtracting, if any, must be less than the number of kids, or else each kid could have received more video games. 5. BRING DOWN: This final step signals the end of dividing up one of the loot piles and represents moving to the next loot pile to be divided up.
Making Math Real further supports successful processing by incrementally guiding students through a four-stage, multisensory learning progression: concrete, semi-concrete, semi-abstract, and abstract. There are three principal modalities for multisensory learning: visual, auditory, and tactile/kinesthetic (motoric). To be multisensory, all three modalities must be engaged and linked all of the time. It is not sufficient that students merely see math problems, hear about them, and then interact with some manipulatives. Rather, the essence is to present incoming information and experience in such organized, incremental, structured, and systematic fashion that all three modalities are fully linked. As a result, the images students see, the language they hear, and the manner in which they interact with manipulatives are all interconnected.
The concrete stage uses hands-on manipulatives, color-coding linked to place value, and informal language to create clear and concrete conceptual images of all math content. The language and the physical movements of the manipulatives are chunked into small, simplified, manageable parts. The language (auditory), the picture imagery (visual) and the manipulatives (motoric) are carefully linked to such a degree that if one of the modalities were “turned off,” say the visual, the language and the interactions with the manipulatives would recreate the same picture imagery. The overall experience is entirely concrete. The symbolic is introduced as a means to record the concrete experience only. Therefore, at the purely concrete level, students understand that the concrete picture of math is replicated by the symbolic one because they are one in the same. The symbolic and the real (concrete) pictures tell the same story.
The semi-concrete stage has students perform a problem at the concrete level and then repeat the same problem from the color-coded math prompt only. The purpose of this level is to help students begin to see that the symbols of math, directly related to the manipulatives, can be enough to recreate the concrete picture and story of the math problem. The semi-abstract stage furthers this development by having students do math problems without the use of manipulatives. However, the problems are still color-coded by place value as a cueing system to help students make the connection that the symbols of math recreate the concrete experience of using the manipulatives. Synthesis at the abstract level, the final stage, occurs when students can look at a problem in their math books and find meaning in the symbols. At this level students independently know and understand what to do. The successful experience of going through the developmental progression of concrete through the abstract helps students integrate concepts with procedures while building and strengthening basic skills.
Students who count on their fingers are showing that they cannot see sequences of numbers in their minds. They make physical attempts to keep track of their counting as they touch parts of their bodies or make tally marks. A requisite brain tool for seeing numbers in the mind is symbol imaging. Making Math Real emphasizes the development of symbol imaging in every game, activity and lesson. A principal method for developing symbol imaging has students focus on sequences of symbols such as the multiplication facts for the fours, and then take mental pictures of them by flooding the three modalities (visual, auditory, and motoric). As students are watching, the teacher erases a specific product, one at a time. Students are required to reconstruct the multiplication facts including the product(s) that have been erased. Eventually, all of the products have been erased and students are able to mentally reconstruct the missing products in sequential or random order because they are able to hold the symbols in mind. Students are required to reconstruct the multiplication facts in a variety of daily, directed multisensory games and activities. The repeated practice of re-imaging, reconstructing, and retrieving the facts helps develop symbol imaging so that students are increasingly able to hold longer and longer sequences of numbers and symbols. The successful development of symbol imaging is one of the main components for getting students off their fingers and learning their math facts.
It is unfortunate that so many students are suffering unnecessarily from math anxiety, especially since most of them have the intelligence, motivation, and math ability to be successful. Math should be the easiest subject to teach since it is entirely concrete, and therefore allows for the systematic guidance from concrete to abstract. Using multisensory structured methods to create real story-based visual images helps students make the essential connections between concepts and procedures. Nothing succeeds like success. Students make the connections, their anxiety diminishes and they are empowered by the realization that they can do math, too.