**Research Basis of Conceptua Math. **

Research in math pedagogy, student cognition and motivation of learning are the building blocks that form Conceptua's DNA. From research about which visual models work best, to teacher training and student motivation, Conceptua Math makes possible the best practice that researchers have brought to light.

- Visual Models for Deep Understanding
- Enhancing Content and Pedagogical Knowledge for Teachers
- Classroom Discussions and Lesson Structure
- Concrete-Representational-Abstract Instructional Approach (C-R-A)
- Intrinsic Motivation
- Journaling in Mathematics
- Real World Investigations and Authentic Data
- Differentiation
- Universal Design for Learning (UDL)

Visual modeling in mathematics is at the heart of Conceptua Math. Our process is to research the finest, most proven visual models and render them in dynamic, interactive formats for both students and teachers to gain deep understanding of mathematical concepts. Our designers, engineers, and professional development experts all read the research on visual models and interact directly with some of the leading researchers.

Upon founding our company, we were influenced by *Adding It Up*, the work of John Van De Walle, and we enjoyed a close collaboration with Dr. Julie McNamara. We have thoroughly studied the *NCTM Curriculum Focal Points* books (cited below) and have enjoyed the opportunity to work with Dr. Karen Fuson. We have also received input on models from other leading researchers including Dr. Nadine Bezuk, Dr. Judith Sowder, and Dr. Barbara Dougherty.

- Fuson, K., SanGiovanni, J., Adams, T., Beckmann, S. (Ed.) (2009). Teaching with Curriculum Focal Points - Focus in Grade 5. Reston, VA National Council of Teachers of Mathematics
- Kilpatrick, J., Swafford, J., and Findell, B. (Eds.). (2001). Adding It Up: Helping Children Learn Mathematics. National Research Council. Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington: National Academy Press.
- McNamara, J., Shaughnessy, M.M. (2010) Beyond Pizzas & Pies: 10 Essential Strategies for Supporting Fraction Sense, Grades 3-5. Sausalito, CA. Math Solutions.
- Schielack, J. (Ed.) (2009). Teaching with Curriculum Focal Points - Focus in Grade 3. Reston, VA National Council of Teachers of Mathematics
- Schielack, J. (Ed.) (2009). Teaching with Curriculum Focal Points - Focus in Grade 4. Reston, VA National Council of Teachers of Mathematics
- Van De Walle, John (2001). Elementary and Middle School Mathematics: Teaching Developmentally. New York: Addison Wesley Longman.

As a publisher, we assume the responsibility to enable successful TEACHING along with successful learning. We observe that technology is sometimes used to bypass the teacher and circumvent excellent classroom dynamics. We rely upon the research of such luminaries as Dr. Deborah Ball and Liping Ma, who of both helped us to understand the content and pedagogical requirements for elementary and middle school mathematics teachers. We find that the arguments about building teacher capacity noted in the National Mathematics Advisory Panel report of 2008 make a very strong case.

- Ball, D. L., Ferrini-Mundy, J., Kilpatrick, J., Milgram, R. J., Schmid, W., Schaar, R. (October 2005) Reaching for common ground in k-12 mathematics education, Notices of the AMS, Vol. 52, No. 9, pp. 1055-1058.
- Ball, D. L., S., and Mewborn, D. (2001) Research on teaching mathematics: The unsolved problem of teachers? mathematical knowledge. In V. Richardson (Ed.), Handbook of research on teaching (4th ed.). New York: Macmillan
- Ball, D. L., Hill, H, C, and Bass H. (2005) Knowing Mathematics for Teaching: Who Knows Mathematics Well Enough To Teach Third Grade and How Can We Decide?. American Educator (pp. 14-46)
- Ma, Liping (1999). Knowing and Teaching Elementary Mathematics. Mahwah, New Jersey, Lawrence Erlbaum Associates.
- National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. Retrieved April 20, 2008, from the U.S. Department of Education Web site: http://www.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf
- Rowan, B., Schilling, S. G., Ball, D. L. and Miller, R. (2001), Measuring teachers? pedagogical content knowledge in surveys: An exploratory study. Consortium for Policy Research in Education, Study of Instructional Improvement, Research Note S-2. Ann Arbor: University of Michigan.
- Behr, M. & Post, T. (1992). Teaching rational number and decimal concepts. In T. Post (Ed.), Teaching mathematics in grades K-8: Research-based methods (2nd ed.) (pp. 201-248). Boston: Allyn and Bacon.
- Bezuk, N., & Cramer, K. (1989). Teaching About Fractions: What, When, and How? In P. Trafton (Ed.), National Council of Teachers of Mathematics 1989 Yearbook: New Directions For Elementary School Mathematics (pp. 156-167). Reston, VA: National Council of Teachers of Mathematics.
- Cramer, K., Behr, M., Post T., Lesh, R., (2009) Rational Number Project: Initial Fraction Ideas. Originally published in 1997 as Rational Number Project: Fraction Lessons for the Middle Grades - Level 1, Kendall/Hunt Publishing Co., Dubuque Iowa.
- Cramer, K., Henry, A., (2002) Using Manipulative Models to Build Number Sense for Addition of Fractions. National Council of Teachers of Mathematics 2002 Yearbook: Making Sense of Fractions, Ratios, and Proportions (pp. 41-48). Reston, VA: National Council of Teachers of Mathematics

Our CEO, Arjan Khalsa, worked with the California Mathematics Project in the 1980s at a time when the project was encouraging deep discussion in elementary school math classes and many teachers and administrators thought the suggestion to be incomprehensible. Now, some 30 years later, rich discussion in the mathematics class is widely regarded as being very important. Our content developers are all versed in the Classroom Discussions book from Math Solutions and the 5 Practices book from NCTM. The work of Michael Schmoker serves as a template for the organizational principles of beginning the class period with an Opener and closing the period with a Closer.

- Chapin, Canavan and Anderson, Catherine O’Connor. (2003) Classroom Discussions: Using Math Talk to Help Students Learn. Sausalito, CA. Math Solutions.
- Herbel-Eisenmann, B. & Cirillo, M. (2009) Promoting purposeful discourse. Reston, VA: NCTM
- Huinker, D., Freckmann, J. L. (2004) Focusing conversations to promote teacher thinking. Teaching Children Mathematics, 1- (7) 352-357
- Schmoker , Michael. (2006) Results Now: How We Can Achieve Unprecedented Improvements in Teaching and Learning. Association for Supervision and Curriculum Development. Alexandria, VA.
- Small, M. (2009) Good Questions: Great Ways to Differentiate Mathematics Instruction
- Smith, M. and Stein, M. (2011) 5 Practices for Orchestrating Productive Mathematics Discussions. Reston, VA National Council of Teachers of Mathematics.
- Hintz, Allison and Kazemi, Elham. (2014) Intentional Talk: How to Structure and Lead Productive Mathematical Discussions

Two of our founders, Arjan Khalsa and Lauri Susi, have both used the C-R-A method in mathematics education as teachers in classrooms and as faculty in training teachers dating back many years. Recently, this methodology has become associated with Singapore Math and the Common Core movement in mathematics. Some of the representative sources are listed below. In practice, much of the input on C-R-A methodology is coming from practical usage out of Singapore. We pay close attention to the work of Ban Har Yeap and others who propose carefully crafted problem types that lend themselves to this sequential approach.

- Bruner , Jerome. (1990) Acts of Meaning: Four Lectures on Mind and Culture. Cambridge, MA. Harvard University Press.
- Carpenter, Thomas P. Children's Mathematics: Cognitively Guided Instruction. Portsmouth, NH: Heinemann, 1999.
- Concrete-Representational-Abstract Instructional Approach Summary Report. The Access Center, American Institutes for Research, Washington, DC. http://bit.ly/1ASeYKT
- Empson, Susan B., and Linda Levi. Extending Children's Mathematics: Fractions and Decimals. Portsmouth, NH: Heinemann, 2011.
- Miller, S. P., Mercer, C.D., & Dillon, A. S. (1992) CSA: Acquiring and Retraining Math Skills. Intervention in School and Clinic, 28, 105-110.
- Mercer, C.D., & Mercer A. S. (2005). Teaching Students with Learning Problems (7th ed). Upper Saddle River, NJ: Pearson Prentice Hall

The highest forms of motivation are intrinsic. With regard to motivation, the research basis for Conceptua Math is about creating products and interactions that lead to successful learning experiences for students and teachers together. We are fascinated by the work of luminaries like Carol Dweck and Daniel Pink who help us all understand that humans innately want to learn, and get a great deal of satisfaction from doing so.

- Dweck , Carol S. (2008) Mindset: The New Psychology of Success. Ballantine Books. New York, NY.
- Pink, Daniel. (2009) Drive: The Surprising Truth About What Motivates Us. Riverhead Books. New York, NY.

The research that we have read and in which we have directly engaged indicates that students should write about their mathematical experiences on a regular basis. When our CEO, Arjan Khalsa, encouraged writing in the mathematics curriculum as part of the California Mathematics Project in the 1980s, some teachers found the idea to be completely outlandish. Today, the practice is more widely accepted, and Conceptua Math includes daily prompts for journaling.

- Leinwand, Steven. (2009) Accessible Mathematics: 10 Instructional Shifts That Raise Student Achievement. Heinemann. Portsmouth, NH.
- Pugalee, David K. (2005) Writing To Develop Mathematical Understanding. Christopher-Gordon Publishers, Inc. Norwood, MA.

It is wonderful when a child becomes deeply engaged in a short story or novel, where the experiences in the text connect to the youngster’s personal life. It is equally wonderful when a child comes to understand how mathematical thinking connects to meaningful experiences in life. The work of Daniel Pink shows how people are motivated when working with their own, personal data. Steve Leinwand’s book, *Accessible Mathematics*, is one of our favorite treatises on great quality math education and we are particularly moved by his references to authentic data.

- Leinwand, Steven. (2009) Accessible Mathematics: 10 Instructional Shifts That Raise Student Achievement. Heinemann. Portsmouth, NH.
- National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. Retrieved April 20, 2008, from the U.S. Department of Education Web site: http://www.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf
- Pink, Daniel. (2009) Drive: The Surprising Truth About What Motivates Us. Riverhead Books. New York, NY.
- Fosnot, C.,T. & and Dolk, M. Young Mathematicians at Work: Constructing Fractions, Decimals and Percents. Portsmouth: Heinemann, 2002
- Fosnot, C.,T. & and Dolk, M. Young Mathematicians at Work: Constructing Number Sense, Addition and Subtraction. Portsmouth: Heinemann, 2002

Our team has many years of experience teaching in classrooms comprised of students with a wide range of abilities and learning styles, and our work in classrooms and direct research directly influences many features of our solution. We understand the importance of planning for differentiation and believe that best practice in differentiated instruction should be in place in all classrooms to meet the needs of all students.

- Celedon-Pattichis, S., & Ramirez, N. G. (2012). Beyond good teaching : advancing mathematics education for ELLs. Reston, VA: National Council of Teachers of Mathematics.
- Steedly, K. Dragoo, K. Arefeh, S., Luke, S. D. (2008). Effective mathematics instruction, evidence for education. 3(1). Downloaded 2009 from http://nichcy.org/wp-content/uploads/docs/eemath.pdf
- Tomlinson, C. (1995). How to differentiate instruction in the mixed ability classroom. Alexandria, VA: Association for Supervision and Curriculum Development.
- Woodward, J., Rieth, H. Review of Educational Research. Vol. 67, No. 4. (1 January 1997), pp. 503-536,
- Gersten R, Beckman S, Clarke B, Foegen A, Marsh L, Star JR, Witzel B. Assisting students struggling with mathematics: Response to intervention (RtI) for elementary and middle schools (NCEE 2009-4060) Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education; Retrieved Nov 9, 2010 from http://ies.ed.gov/ncess/wwc/publications/practiceguides/

The team of people who founded Conceptua Math have years of direct research experience in special education technology and universal design for learning. Together, we pioneered the work in an assistive technology company called IntelliTools. We find it fascinating that the mathematics community shares fundamental educational principles with the UDL movement: multiple means of representation, multiple means of expression, and multiple means of engagement.

- Rose, D., Hasselbring, T. S., Stahl, S., & Zabala, J. (2005). Assistive technology and universal design for learning: Two sides of the same coin.
- Edyburn, D. , Higgins, K. , & Boone, R. (Eds.), Handbook of special education technology research and practice (pp. 507-518). Whitefish Bay, WI: Knowledge by Design, Inc. Abstract