What Is It and Why Does It Work?

Communicator Mathematics™ is a research-based, comprehensive standards-based, constructivist mathematics program for grades 3 through 6 (and an ancillary program for grades 7 and 8) that includes both math content and pedagogy.

Philosophy

Communicator Mathematics™ is based on the principle that all children can learn sophisticated mathematics and demonstrate strong computational and mental mathematics skills under certain conditions:

• The material must be presented in a way that keeps children interested and actively involved.
• The program must build a foundation of conceptual understanding before algorithms are introduced.
• The design and implementation of the program must be sufficiently comprehensive to make "test prep" irrelevant.

Assumptions:

• Standards-based mathematics requires that teachers have a deep understanding of the mathematical principles underlying the content to be taught.
• Pre-service and in-service programs for elementary teachers generally do not include sufficient mathematical content or appropriate pedagogical theory and practice.
• Instruction needs to address all students’ learning styles.
• Materials must be teacher-friendly.
• An emphasis on the attainment of traditional computational and mental math skills must not be sacrificed to the development of mathematical concepts, or vice versa, but the two elements must complement each other.
• A transition must be provided to lead students from the conceptual development to an internalization of the algorithm.

Content Description:

• Linked to NCTM Standards
• Spiraled to provide for incremental understanding, meaningful repetition, growth and retention
• Developed through hands-on activities
• Stresses both computational fluency and math concepts

Components:

• All instructional materials are included (detailed daily lesson plans, all teaching materials needed to implement each lesson, and class activities that promote active learning)

Foundations in Research

The overall goals of Communicator Mathematics®  are to provide:

• A standards- and research-based mathematics program based on the principle that all children can learn world-class mathematics, specifically,
• Children will learn and develop a deep understanding of the mathematical principles and concepts as described in the National Council of Teachers of Mathematics (NCTM) Principles and Standards for School Mathematics as well as in the specific content standards of the state(s) in which it is implemented.
• Children will learn and be able to apply their mathematical knowledge when needed, including the recall of specific algorithms and facts, such as multiplication facts.

• Comprehensive support for the teacher so that implementation will be effective and the program will achieve its goals, in that
• Teachers who may be math phobic or challenged by the theory and content of standards-based mathematics and pedagogy will be able to provide interesting and intellectually challenging lessons that meet the needs of their students.
• Teachers are differentiated learners too; the Communicator Mathematics®  program is appropriate for teachers at all points on the learning curve for excellence in instruction.

A Selected Bibliography

This information describing the research base for Communicator Mathematics® is organized in relationship to several principles that formulate the general foundation of the program.  This document provides an overview of some of the research that addresses each of these principles and how they are applied. In addition, the cited research often informs other, additional points that may be discussed in other sections of this paper.  Some of the sources cited provide support for changes made for the 2008 revision.

Communicator Mathematics® has the advantage of learning from the groundbreaking programs that have gone before it.  It stands on the shoulders of giants and has tried to benefit from their experience.

Principle A:
All children can learn. Those children who learn world-class mathematics have a comprehensive and deep understanding of both the specific mathematics being  learned and of the underlying mathematical principles and the inter-related nature of various topics within that field.

As seen in Communicator Mathematics®

A review of the planning guides for Communicator Mathematics®  reveals that approximately five to seven different activities are included in every lesson. These activities represent the five major strands in mathematics to ensure that students are consistently investigating and are involved in experiencing theholistic approach to mathematics  as well as making connections among these strands. In essence, students who experience Communicator Mathematics® do not learn skills and concepts in a disjointed and isolated manner.

In addition, active participation is built into every lesson.  Through the use of the CommunicatorÒ clearboard, each student is involved with every question and idea under consideration.  Not only is the teacher consistently informed about how well focused every student is on every topic, he or she also knows how well the student is processing and understanding the material.

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Ball, D. and H. Bass.  “Making Believe:  The Collective Construction of Public Mathematical Knowledge in the            Elementary Classroom.” In Constructivism in Education. D. Phillips (Ed.).(2000) Chicago: 
           University of Chicago Press. 193-224.

Bell, M. S. “Mathematical Uses and Models in Our Everyday World.” Studies in Mathematics,
           Volume XX.  Stanford:  School Mathematics Study Group. (1972) Eric Ed 143-557.

———. “What does ‘Everyman’ Really Need from School Mathematics?” Mathematics Teacher.
           no 67 (1974): 196-202.

Bransford, John D., Ann L. Brown and Rodney R. Cocking (Eds.).  How People Learn: Brain, Mind,
           Experience and School.(2003) Washington D.C.: National Academy Press.

Carpenter, T. P. “Conceptual Knowledge as a Foundation for Procedural Knowledge.” In Conceptual 
            and Procedural Knowledge:  The Case of Mathematics, edited by J. Hiebert. (1984) 113-132.
            Hillsdale, NJ: Erlbaum.

Hiebert, James.  “Children’s Mathematics Learning: The Struggle to Link Form and Understanding.”
            Elementary School Journal, 84 (1984) 497-513.

———. “What Research Says About the NCTM Standards.” In A Research Companion to
            Principles and Standards in School Mathematics. (2003) 5 – 23. Reston, VA: National
           Council of Teachers of Mathematics.

Isaacs, A. C., W. M Carroll, and M. Bell.  A Research-Based Curriculum:  The Research Foundations of the            UCSMP Everyday Mathematics Curriculum. (1998) Chicago, IL: UCSMP.

Kilpatrick, J.,  J. Swafford and B. Findell (Eds.) Adding It Up: Helping Children Learn Mathematics.
           (2001) Washington, D.C.: National Academies Press.  

Madsen, Anne L. and Perry Lanier. “Does Conceptually Oriented Instruction Enhance Computational
           Competence?” Focus on Learning Problems in Mathematics. 17, (Fall, 1995) 42-64.

National Council of Teachers of Mathematics (NCTM). Principles and Standards for School
            Mathematics.  (2000) Reston, VA: NCTM.

Pollak, H. O. “The Mathematical Sciences Curriculum K-12: What is Still Fundamental and What is
            Not.” Reprinted in The Monitoring of School Mathematics: Background Papers. T. A. Romberg
            and D. M. Stewart (Eds.) (1987) Madison, WI: Wisconsin Center for Educational Research.
           117-133.

Stigler, J. W. and M. Perry. “Mathematics Learning in Japanese, Chinese, and American classrooms.” In            Children’s Mathematics: New Directions for Child Development. G. B. Saxe and M. Gearhart (Eds.)
           (1988) San Francisco: Jossey Bass.

Williams, W. M., et. al. Practical Intelligence for School.  (1996) New York: Harper Collins College
           Publishers.

Woodward, J. and J. Baxter.  “The Effects of an Innovative Approach to Mathematics on Academically
            Low-achieving Students in Inclusive Settings.” Exceptional Children. 63 (1997) 373-388.

Principle B:
Children learn effectively when they are taught mathematics through problem solving, and appropriate technology can be used to enhance the teaching of problem solving.

As seen in Communicator Mathematics®

“Mathematics is a way of thinking” (Bergeron and Herscovics, 1990).

All concepts taught in Communicator Mathematics® progress from hands-on, discovery-based lessons to iconic representations and finally to symbolic representation. Students consistently create sets of “data” through investigations.  Then, through developing conjectures and questioning them, they discover and/or create a related algorithm. Procedures and algorithms are not stressed, applied or  practiced until conceptual understanding is established.

Throughout the program, teachers are provided with suggestions and guidance so that they have examples of questioning that leads students to move from the basic levels of cognition into higher-order thinking appropriate to the topic.  When presented properly, children of all ages are able to think at the analysis, synthesis and evaluation levels of cognition.

The use of appropriate calculators is integrated throughout the program, and thorough directions are provided for the teacher so that students use these devices as a tool for presenting data to be considered and interpreted, not just a means of doing calculations. 

_______________

Bergeron, J. C. and N. Herscovics. “Psychological Aspects of Learning Early Arithmetic.” In
            Mathematics and Cognition: A Research Synthesis by the International Group for the Psychology of Mathematics Education. P. Nesher and J Kilpatrick (Eds.). (1990) Cambridge, England:
            Cambridge University Press.

Bell, M. S. “Calculators in Elementary Schools: Some Tentative Guidelines and Questions Based on
            Classroom Experience.” Arithmetic Teacher. 23 (1976) 502-509.

Ben-Hur, Meir. Concept Rich Mathematics Instruction:  Building a Strong Foundation for Reasoning and
           Problem Solving. (2006) Alexandria, VA: Association for Supervision and Curriculum Development.

Bruner, J. Acts of Meaning. (1990) Cambridge, MA: Harvard University Press.

Cuevas, G. and K. Yeatts.  Navigating Through Algebra in Grades 3-5. (2001) Reston, VA: NCTM.

Dewey, J. How We Think, Revised Ed. (1933/1998) Boston, MA:  Houghton Mifflin Company.

Fosnot, Catherine Twomey and Maarten Dolk.  Young Mathematicians at Work: Constructing Number
            Sense, Addition and Subtraction. (2001) Portsmouth, NH: Heinemann.

Donovan, M. S. and J. D. Bransford. (Eds.). How Students Learn: History, Mathematics, and Science in
            the Classroom. (2005) Washington D.C.: National Academies Press.

Friel, S., S. Rachlin and D. Doyle. Navigating Through Algebra in Grades 6-8. (2001) Reston, VA:
            NCTM.

Felder, R. M. and R. Brent. “Learning by Doing.” Chemical Engineering Education. 37(4) (Fall, 2003)
            282-283.

Gardner, H. and T. Hatch. “Multiple Intelligences go to School:  Educational Implications of the Theory
            of Multiple Intelligences.”  Educational Researcher. 18(8) (1993) 4-9.

Hiebert, James.  “What Research Says About the NCTM Standards.” In A Research Companion to
            Principles and Standards in School Mathematics. (2003) 5 – 23. Reston, VA: National Council of
           Teachers of Mathematics.

Kilpatrick, Jeremy, Jane Swafford, and Bradford Findel (Eds.). “Mathematics Learning Study Committee,
           National Research Council , Conclusions and Recommendations” In Adding It Up: Helping
           Children Learn Mathematics. Washington, D.C.: The National Academies Press. 407-432.

Lesh, R., T. Post and M. Behr. “Representations and Translations Among Representations in Mathematics            Learning and Problem Solving.” In Problems of Representation in the Teaching and Learning
           of Mathematics. R. J. Sternberg (Ed.) (1987) Hillsdale, NJ: Erlbaum. 33-40.

Murray, Miki with Jenny Jorgensen. The Differentiated Math Classroom: A Guide for Teachers, K-8.
            (2007) Portsmouth, NJ: Heinneman.

Piaget, J. The Psychology of the Child. (1972) New York: Basic Books.

Sousa, David A. How the Brain Learns, Third Edition. (2005) Thousand Oaks, CA: Corwin Press, a
            division of Sage Publications Co.

———. How the Brain Learns Mathematics. (2008) Thousand Oaks, CA: Corwin Press, a
            division of Sage Publications Co.

Usiskin, Z. and M. S. Bell.  Applying Arithmetic:  A Handbook of Applications of Arithmetic. (1983)
            Chicago:  The University of Chicago.  ERIC SE 046 244, SE 046 245, SE 046 246.

Vygotsky, L. Mind in Society. (1978) Cambridge, MA: Harvard University Press.

Principle C:
Retention of knowledge, concepts and skills is most effective when both the teaching and practice are distributed over a period of time.

As seen in Communicator Mathematics®

A review of the planning guides in Communicator Mathematics® reveals that students experience well- sequenced activities relating to the same skill or concept over a period of  four to five days, each time repeated but with additional information, focus and more advanced activities.  All homework activities provide short, interesting practice of the concepts being taught. Games and other activities not only provide interesting, appropriate practice, but are elements of the program that help the teacher differentiate instruction when needed.

This structure is especially important in considering the underlying approach used in building computational fluency.  Through carefully designed spiraling, students are not only consistently presented with alternative strategies to determine answers but are also taught when mental math, paper and pencil or estimation and a calculator are appropriate strategies for the task.

_______________

Caple, C. The Effects of Spaced Practice and Spaced Review on Recall and Retention Using
           Computer Assisted Instruction. (1996) Ann Arbor, MI: UMI.

Cook, C.J. and J. A. Dossey. “Basic Fact Thinking Strategies for Multiplication – Revisited.” Journal for
            Research in Mathematics Education. 13 (1982) 163-171.

Kilpatrick, Jeremy, Jane Swafford, and Bradford Findel (Eds.). “Mathematics Learning Study Committee,
           National Research Council , Conclusions and Recommendations” In Adding It Up: Helping
           Children Learn Mathematics. Washington, D.C.: The National Academies Press. 407-432.

Pesek, Dolores D. and David Kirshner. “Interference of Instrumental Instruction in Subsequent Regional            Learning.” Journal for Research in Mathematics Education. 31 (5). 524-540.

Rea, Cornelius P. and Vito Modigliani.  “The Effect of Expanded Versus Massed Practice on the
            Retention of Multiplication Facts and Spelling Lists.” Human Learning. 4 (1985) 11-18.

Renner, J. et. al. Research, Teaching and Learning With the Piaget Model. (1976) Norman, OK:
            University of Oklahoma Press.

Willingham, Daniel. Allocating Student Study Time: ‘Massed’ versus ‘distributed’ practice. American
           Educator. Summer, 2002.

Principle D:
Well-designed assessments and assessment practices have a positive effect on
both teaching and learning.

As seen in Communicator Mathematics®

Communicator Mathematics® includes both formal and informal as well as formative and summative assessments. 

The Communicator® clearboard is a tool that permits teachers to instantly assess every student’s understanding of the concepts being presented in a lesson It is often used along with the Resources for Active Learning and Discovery Templates materials that are provided in lieu of  those typically designed by the teacher for class work..  These program elements help teachers modify and adjust instruction to meet the needs of the students on an on-going basis throughout the lesson.  

Both shorter and longer assessments, including those for mental math, are spaced at regular intervals throughout the program.  The more traditional assessment instruments are designed in the formats most frequently seen  on high-stakes tests.  In this way “test prep” is built into the program, marrying quality instruction to quality assessment, and vice versa.  Thus, when high-stakes assessments are given, students will not be confused by the format or question design of the tests. They are able to show what they know and will have developed a sense of pacing and stamina that supports their ability to perform.  Finally, throughout the year, parents, teachers, supervisors and administrators have a way of knowing how both individual students and groups of students are progressing.

Not only are the assessments designed to reflect the mathematical content of the program, but also to support and extend the emphasis on thinking skills and problem solving. Because writing and higher-level thinking activities are an integral part of  class activities, homework investigations, and assessments, students are consistently required to provide reasons and explanations of how they determined solutions or how they reached various mathematical conclusions.

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Bell, M. S. and J. B. Bell.  Assessing and enhancing the Counting and Numeration Capabilities and Basic            Operation Concepts of Primary School Children. (1988) Chicago:  University of Chicago.

Black, P. and D. William. “Assessment and Classroom Learning.” Assessment in Education, 5 (1): (1998) 7-74.

———. “Inside the Black Box: Raising Standards Through Classroom Assessment.” Phi Delta Kappan, 80 (2):            (1998) 139-148.

Checkley, Kathy. The Essentials of Mathematics K-6:  Effective Curriculum, Instruction and Assessment.
           (2006) Alexandria, VA: Association of Supervision and Curriculum Development.

Cobb, P., et. al. “Assessment of a Problem Centered Second-Grade Mathematics Project.”  Journal for
            Research in Mathematics Education. 22(1) (1991) 3-29.

Principle E:
Development of traditional mathematical skills, particularly computation, mental math and math facts remains important. So-called “reform” approaches to teaching mathematics have been shown to have wide-ranging applications and to be more effective in achieving the goals of world-class education than more traditional approaches.  Nevertheless, there continues to be a need for children to learn certain traditional math such as those inherent in multiplication and standard algorithms, and to be able to apply appropriate strategies for computation and the use of technology.

As seen in Communicator Mathematics®

Communicator Mathematics® includes a mental math activity in over ninety percent of the lessons. Every six days students are given a mental math assessment.  On some occasions, these assessments evaluate a particular skill, but on other occasions, the skills are combined and students must quickly decide which operation is appropriate.  The mental math skills are integrated into the regular daily activities as well.

Another important example relates to multiplication.  After multiplication is investigated, and the concept of what multiplication is has been established, facts are consistently practiced not only through traditional recall but also through conceptual application. Students are made metacognitively aware of what the skill is and how to perform the calculation. Even with fractions, students are expected to use mental math techniques to complete computations that involve fractions with compatible denominators.

_______________

Baroody, A. J. and H. P. Ginsburg.  “The Relationship between Initial Meaning and Mechanical
            Knowledge of Arithmetic.” In Conceptual and Procedural Knowledge:  The Case of
            Mathematics. J. Hiebert (Ed.) (1986) Hillsdale, NJ: Erlbaum.

Carpenter, T. P. and J. M. Moser. “The Acquisition of Addition and Subtraction Concepts in Grades
           One Through Three.” Journal for Research in Mathematics Education.  15 (1984) 179-202.

Fennimore, T. and M. Tinzmann. What is a Thinking Curriculum? (1990) Oak Brook, IL: North Central
            Regional Educational Laboratory.

Hiebert, James.  “What Research Says About the NCTM Standards.” In A Research Companion to
            Principles and Standards in School Mathematics. (2003) 5 – 23. Reston, VA: National Council
            of Teachers of Mathematics.

Isaacs, A. and W. M. Carroll. “Strategies for Basic Facts Instruction.” Teaching Children Mathematics.
            88(4) (1995) 276-279.

Kilpatrick, J. “What Works?” In Standards-Based School Mathematics Curricula:  What Are They? What Do             Students Learn?”  S. Senk and D. Thompson (Eds.) (2003) Mahwah, NJ: Lawrence Erlbaum
            Associates.

Schoenfield, Alan H.  “Making Mathematics Work for All Children:  Issues of Standards, Testing, and
           Equity.” Educational Researcher. 31 (1): (2002) 13-25.

Stigler, J. W.,  K. C. Fuson, M. Ham, and M. S. Kim. “An Analysis of Addition and Subtraction Word
            Problems in American and Soviet Elementary Mathematics Textbooks.” Cognition andInstruction.
            3 (1986) 153-171.

——— and J. Hiebert. “Understanding and Improving Classroom Mathematics Instruction: and
            Overview of the TIMMS Video Study.” Phi Delta Kappan, 79(1). (1997). 14-21.

Walsh, D. J. “Extending the Discourse on Developmental Appropriateness:  A Developmental
            Perspective.”  Early Education and Development. 2 (19991) 109-119.

Principle F:
Appropriate professional development and other support systems are an
essential part of reform in mathematics education.

As seen in Communicator Mathematics®

It cannot be assumed that every teacher is well prepared to teach standards-based, hands-on, discovery-oriented, problem-solving lessons even with well-prepared, research-based materials.  Professional development support is built into Communicator Mathematics®.

On the most basic level, Communicator Mathematics® provides support for teachers through detailed lesson plans and video vignettes as well as providing all the materials for class work, homework and assessments, leaving the teacher free to prepare for teaching. The lesson plans not only make teaching suggestions, they also provide directions regarding such things as how to handle manipulatives and suggest exploratory questions.  They also give detailed information about the mathematical principles that underlie the concepts to be taught and emphasize teaching strategies for concepts and skills that are generally troublesome for students.  Since standards-based mathematics at the elementary level includes foundational material for more advanced math, these structural elements are very critical to the long-range math achievement of the students.  All lesson plans and materials are made available over the internet, so that the teacher has easy access to them from almost any location.

Coaching, which includes both traditional workshops and in-class demonstration lessons, co-teaching and more traditional “coaching” of the teacher’s performance are important elements of a comprehensive implementation of the Communicator Mathematics® program.  Full day workshops, scheduled at regular intervals throughout the year, act as an overview of the upcoming materials and provide instruction in the mathematical concepts to be taught and the underlying rationale for the various activities that will be utilized.  Finally the presenters demonstrate the pedagogical techniques to be employed and model effective teaching practices.

At the same time, practical, interesting, detailed instruction in the theory and practice of teaching standards-based mathematics at the elementary and middle school levels is available through on-line courses.

The workshop presentations are followed by demonstration lessons presented by the coach in the teacher’s classroom. Generally the coach will have a brief pre-conference before the demonstration so that the coach will be implementing an appropriate lesson and the teacher will be prepared to observe the demonstration. The lessons are followed by a debriefing session which assists the teacher in processing what they have seen. It gives them the opportunity to ask questions and explore issues of concern or interest in the most secure environment possible. The coach is also able to address many of the concerns, such as management of behavior and materials, that teachers have as the focus of instruction moves away from the teacher-centered class to a student-centered one.

As the coaching experience progresses, teachers are encouraged to participate in co-planned, co-teaching sessions.  Again, within this secure environment, paced to meet the individual teacher’s needs, this allows the individual to “try out” various ideas.

On a regular day-to-day basis when the coach will not be in the classroom, instructional support is provided through video vignettes that show all techniques and approaches to all concepts for each lesson on DVD as well as the Internet.  This vehicle reflects the same techniques covered in the workshops, but extends the instruction as a refresher for the teacher. Here, the teachers are able to again see the techniques applied in good practice.

Available via cell phone as well as their regular visits to the classroom, coaches also provide guidance in related areas such as interpreting assessment results, differentiating instruction and dealing with individual student needs.

_______________

Arron, D. Classroom Implementation and Impact of Everyday Mathematics, K-3: Teachers’
            Perspectives on Adopting a Reform Mathematics Curriculum. (1993) Chicago: University of
            Chicago.

Center for Applied Research and Educational Innovation.  Charting a New Course:  A Study of the
            Adoption and Implementation of Standards-Based Mathematics Curricula in Eight Minnesota
            School Districts. (2005)  Minneapolis, MN: University of Minnesota, College of Education and
           Human Development.

Felder, R. M. and R. Brent.  “Navigating the Bumpy Road to Student-Centered Instruction.” College
            Teaching. 44(2) *1996) 43-47.

Fosnot, C. T. Enquiring Teachers, Enquiring Learners:  A Constructivist Approach to Teaching. (1998)
            New York: Teachers College Press.

Herbel-Eisenmann, Beth A. “From Intended Curriculum to Written Curriculum: Examining the ‘Voice’ of
            a Mathematics Textbook. Journal of Research in Mathematics Education. 38(4) (July, 2007)
           344-369.

Holt-Reynolds, D. “What Does the Teacher Do? Constructivist Pedagogies and Prospective Teachers’
            Beliefs About the Role of a Teacher.” Teaching and Teacher Education. 16(1) (2000) 21-32.

Jacobs, Victoria R. “Professional Development Focused on Children’s Algebraic Reasoning in
            Elementary School.” Journal for Research in Mathematics Education. 38(3) (May, 2007)
           258-288.

National Council of Teachers of Mathematics. Illuminations. (2005) Reston, VA: NCTM.

North Central Regional Educational Laboratory.  “Critical Issue:  Mathematics Education in the Era of
            NCLB—Principles and Standards” (2005) Washington, D,C.: Learning Point Associates.
            (http://www.ncrel.org/sdrs/areas/issues/content/contareas/math/ma500.htm).

Sosniak, L. “Professional and Subject Matter Knowledge for Teacher Education.” In The Education of
            Teachers. G. Griffin (Ed.) (1999) Chicago: University of Chicago Press.

Principle G.  
The standards and high-stakes assessments vary somewhat state to state.

As seen in Communicator Mathematics®

While Communicator Mathematics® was originally based primarily on the NCTM Standards, LLTeach Inc. keeps itself informed about the standards as they are expressed and implemented in individual states so that its materials can be effectively implemented in any location.

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Council of Chief State School Officers.  Curriculum Frameworks in Mathematics and Science: 
           How Are They Changing Across the States? (1995) Washington, D.C.: CCSSO.