Developing Automaticity with the Multiplication and Division Facts: The 9 Lines Multiplication Fact Acquisition & Application Strategy
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
All math content presented in the elementary grades is intended to provide the specific development and tools necessary to support successful algebraic processing in middle school and high school. Over the previous 34 years I have worked with well over 10,000 students of all ages and processing styles, and in my experience, the most valuable math tool students acquire in the elementary grades is the development of automaticity with the multiplication and division facts. The numerous and varied essential applications and interrelationships of the multiplication and division facts drive most mathematical processing and provide the essential foundations for future problem solving across all grades. All students, through calculus, require fluent access to all multiplication and division fact family configurations in support of:
- Multiplication and division problem solving
- All components of fraction, ratio, and proportion problem solving
- Factoring whole numbers, polynomials, and expressions of higher degree
- Matrices and solving systems of equations
- Simplifying expressions and solving equations in one, two, and more variables
The above list represents only the smallest fraction of math content wherein automaticity of the multiplication and division facts greatly supports successful algebraic processing, because all math content from pre-algebra through calculus is rooted in connections and interrelationships of the multiplication and division facts. However, it is important to note that learning the times tables is far more than only memorizing the factor times factor configurations. The most valuable development in learning the times tables is applying the multiplication facts to:
- Solve for missing factors
- Find greatest common factors and divisors of two products
- Generate least common multiples/denominators of two products
- Find quotients and solve for missing divisors and dividends
- Connect all relationships of the multiplication and division fact families
Unfortunately, it is conservatively estimated that 50% of students nationwide have not developed automaticity with the multiplication and division facts. The struggles educators and students have faced in attempting to teach and learn the multiplication and division facts date back to the beginnings of formal instruction in mathematics. The frustration, confusion, and discouragement educators and students experience when students, regardless of everybody’s efforts, cannot recall the times tables, represents the greatest educational challenge in the entire K-12 math continuum. Not understanding the cognitive development necessary to support automaticity, educators have tried everything from reward motivators to punishment systems, ineffective drill, timed speed tests, flash cards, mnemonic devices, using songs such as the multiplication rock and/or rap, finger tricks and strategies, to name only a few. These methods have not been successful because they do not address the developmental basis of the problem: until students have developed sufficient sensory-cognitive tools supporting access to symbolic memory, they will not be able to image, store or retrieve all of the multiplication or division facts with automaticity. Therefore, all math educators and their students need a proven, comprehensive, developmental, and multisensory structured system for developing automaticity with the multiplication and division facts.
Automaticity refers to the consolidation of mental fluency in computing math facts. Being automatic with the multiplication facts, for example, means students can mentally generate the solution to any of the 100 multiplication facts in any combination without having to count to arrive at the solution. The 100 multiplication facts are all of the factor times factor combinations from 0 x 0 = through 9 x 9 =. In this manner, students know instantly that 6 x 7 = 42 without counting on fingers, skip counting, making dots or tallies, adding 6 seven times, etc. Therefore, automaticity with the multiplication facts helps students generate all configurations of the multiplication and division fact families quickly, fluently, and with minimal demand on working memory, thereby maximizing students’ access to the cognitive tools of working memory directly supporting comprehension, accuracy, and integration.
One of the principle sensory-cognitive developments necessary for being automatic with the math facts is symbol imaging, the visuo-perceptual ability to perceive, hold, store, and retrieve sequences of numbers and/or mathematical symbols. Sensory-cognitive tools, such as symbol imaging, enable us to express what we know – they provide a direct conduit in both directions connecting processing to intelligence. Sensory-cognitive development for math refers to the specific ability of using the visual, auditory, and kinesthetic-motoric senses to engage and support the successful central processing of numerical and/or mathematical symbols. Students with under-developed sensory-cognitive abilities often have limited access to memory and are characteristically challenged by learning, retaining, and applying the math facts, recalling formulas and definitions, remembering the sequences and structure of multi-step problem solving, integrating concepts with their respective procedures, and managing all the details in their procedural work.
I created and designed the 9 Lines Multiplication Fact Acquisition and Application Strategy© primarily to address the development of symbol imaging to support automaticity with the multiplication and division facts. However, the 9 Lines Multiplication Fact Acquisition and Application Strategy is significantly more than learning the multiplication facts. The secondary purpose of the 9 Lines, embedded within its design, is a comprehensive mental organizing system expressly created to support students’ mental fluency in finding products, quotients, missing factors and divisors, greatest common factors and greatest common divisors of two products, and least common denominators (multiples) of two products. The structure, implementation, and applications of the 9 Lines mental organizing system is so extensive that it requires a 3-day course to present. Because the 9 Lines Multiplication Fact Acquisition and Application Strategy provides the foundation for successful math processing across all of the Making Math Real courses, I consider the The 9 Lines Intensive to be the most vital course in support of all the other Making Math Real courses in the series. Furthermore, for the 9 Lines Multiplication Fact Acquisition and Application Strategy to be effective, educators must integrate all elements of its design, structure, implementation, and applications. The 9 Lines strategy is never intended to be used without educators’ full knowledge of how to teach it correctly. As with all multisensory structured methods, every component has been thoroughly considered, tested, and structured for maximum development and success. If any component is omitted, altered, substituted, or incorrectly sequenced, then the outcome will be subverted. For this purpose, The 9 Lines Intensive course provides the following content and structure to ensure all course participants are equipped with sufficient baseline knowledge to begin applying with students:
- Identify the cognitive and affective behaviors indicating students’ level of symbol imaging development
- Learn the multisensory structured method for developing symbol imaging
- Learn all the language prompts and gestures for correctly teaching the 9 Lines
- Learn the sequence of all phases and assessment points
- Teach students to image a new times table
- Develop students’ automaticity across all the tables
- Develop automaticity for the multiplication and division facts simultaneously
- Provide daily maintenance and ongoing prescriptive practice
- Learn the 9 Lines connection to and involvement in teaching the addition facts
- Design prescriptive practice and problem sets
- Provide extensions to games for high-engagement prescriptive practice
- Learn the extensions for whole number factoring
- Learn the 9 Lines mental organizing system to develop mental fluency in finding products, quotients, missing factors and divisors, greatest common factors and greatest common divisors of two products, and least common denominators (multiples) of two products
- Provide extensions for generating lowest terms and equivalent fractions
- Learn to connect whole number factoring to factoring polynomials
Symbol imaging is the primary developmental outcome of the 9 Lines Multiplication Fact Acquisition and Application Strategy, and automaticity with the multiplication and division facts is the practical application of that development. Symbol imaging for numbers is a crucial sensory-cognitive tool, and as educators, we need to assess and address all of our students’ developmental needs to ensure that all students have developed symbol imaging prior to graduation from high school. The development of symbol imaging is not maturational, it requires precise activation combined with sustained daily and prescriptive practice, often for a period of up to three years or more. If educators do not intervene on behalf of students with under-developed symbol imaging, then those students may continue indefinitely into adulthood using inefficient compensatory strategies to solve math facts, and consequently, never fully develop symbol imaging. Therefore, as educators, we must commit to developing our students’ symbol imaging by using interventions specifically designed for that purpose. The use of charts, calculators, or other devices that bypass the direct activation of symbol imaging is highly contraindicated, because once a student emotionally bonds with the chart or calculator, it becomes extremely difficult to get them to let go of that cognitive prosthetic device. If the student resists letting go, then the development of symbol imaging may be significantly protracted or inhibited. Furthermore, for students who become reliant on charts and calculators, these devices become integrated as prosthetics because students must use them at all times to calculate. Frequently, I have observed students of all ages go blank when confronted with a simple math fact such as 2 x 4 =, and reach for the chart or calculator to solve. In addition, continued reliance on devices that do not activate symbol imaging perpetuates a subtractive development where students become less and less able to calculate mentally. In these instances, the chart or calculator supplants students’ abilities to know if solutions are correct, or require correcting, as students accept the calculator’s solutions without the certainty that given solutions are either accurate or reasonable. Lastly, students will be faced with numerous ongoing situations where the use of calculators or charts is prohibited. With their calculating prosthetics removed, these students face almost certain failure, no longer able to accurately express mathematical solutions. The ability to independently calculate mentally is one all of us need and deserve. Therefore, the educational imperative for all math educators is to ensure maximum development of symbol imaging for all students.
The 9 Lines Multiplication Fact Acquisition and Application Strategy is an indispensable resource for all K-community college math educators because it spans and connects all math content as a powerful method for developing symbol imaging and automaticity, while providing the mental organizing structure for all related multiplication and division problem solving. For this purpose, I have intended The 9 Lines Intensive to be the companion course in support of all the courses in the Making Math Real series.