The goal of this study was to build on previous research on Concrete-Representational-Abstract model and Strategic Instruction Model. Concrete-Representational-Abstract is a scaffolding model that begins with a concrete example, moves on to a representational example, and connects those to the abstract concept. In this study, students began using a physical manipulative (foam Base Ten blocks), moved into drawing their own representations of the Base Ten blocks, and connected those to the abstract number sentences. Visual representations, concrete and representational, increase student achievement on abstract concepts.

Flores, M. M., Hinton, V., & Strozier, S. D. (2014). Teaching Subtraction and Multiplication with Regrouping Using the Concrete-Representational-Abstract Sequence and Strategic Instruction Model. Learning Disabilities Research & Practice. 29(2), 75-88. Retrieved from http://onlinelibrary.wiley.com/doi/10.1111/ldrp.12032/abstract

Joan Gujarati's research into helping learners better understand mathematics has been labeled the C-P-A (concrete-pictorial-abstract) method that transfers students from learning mathematics in a concrete way to an abstract way.

This study attempted to maintain controlled conditions between two sets of classrooms. One set of classrooms used physical manipulatives to learn and demonstrate fraction concepts. The other set of classrooms used virtual manipulatives. The study found that students performed just as well, if not better when using the virtual manipulatives.

Mendiburo, M. & Hasselbring, T. (2014). Technology's Impact on Fraction Learning: An experimental comparison of virtual and physical manipulatives. Journal of Computers in Mathematics and Science Teaching, 33(2), 209-231. Chesapeake, VA: Association for the Advancement of Computing in Education (AACE). Retrieved from http://www.editlib.org/p/48044/

In this study, small groups of students with autism were taught mathematics concepts using virtual and physical manipulatives. Students performed equally well using either of the types of manipulatives. However, the students reported that they enjoyed the virtual manipulatives more than physical manipulatives.

Bouck, E. C., Satsangi, R., Doughty, T. T., & Courtney, W. T. (2014). Virtual and Concrete Manipulatives: A Comparison of Approaches for Solving Mathematics Problems for Students with Autism Spectrum Disorder. Journal of Autism and Developmental Disorders. 44(1), 180-193. Retreived from http://link.springer.com/journal/10803

Visualizing Math: Computer Graphics Transforms the Face of Traditional Math Education

Lisa Stapleton’s article “Visualizing Math: Computer Graphics Transforms the Face of Traditional Math Education” discusses how the use of mathematical software helps students visualize math concepts as well as the success the students have. The article also talks about how much Computer-Aided Instruction (CAI)s are improving as well. Some of the software discussed are; Mathematica, a product produced by Wolfram Research Inc., Derive, by Soft Warehouse, f(x), authored by Martin Lapidus of Laxcaux Graphics, Fields & Operators among others.

Stapleton, L. (June, 1990). Visualizing Math: Computer Graphics Transforms the Face of Traditional Math Education. Computer Graphics World, 58. Retrieved from http://www.cgw.com/Publications.aspx

## Research to Support Using Visualizations in Mathematics

## Using Concrete-Representational-Abstract Sequence and Strategic Instruction Model

The goal of this study was to build on previous research on Concrete-Representational-Abstract model and Strategic Instruction Model. Concrete-Representational-Abstract is a scaffolding model that begins with a concrete example, moves on to a representational example, and connects those to the abstract concept. In this study, students began using a physical manipulative (foam Base Ten blocks), moved into drawing their own representations of the Base Ten blocks, and connected those to the abstract number sentences. Visual representations, concrete and representational, increase student achievement on abstract concepts.Flores, M. M., Hinton, V., & Strozier, S. D. (2014). Teaching Subtraction and Multiplication with Regrouping Using the Concrete-Representational-Abstract Sequence and Strategic Instruction Model. Learning Disabilities Research & Practice. 29(2), 75-88. Retrieved from http://onlinelibrary.wiley.com/doi/10.1111/ldrp.12032/abstract

## Deepening Mathematics Teaching and Learning through the Concrete-Pictorial-Abstract Approach

Joan Gujarati's research into helping learners better understand mathematics has been labeled the C-P-A (concrete-pictorial-abstract) method that transfers students from learning mathematics in a concrete way to an abstract way.Gujarati, J. (2013). Deepening Mathematics Teaching and Learning through the Concrete-Pictorial-Abstract Approach. Strategies for Successful Learning, 6. Retrieved from __http://www.ldworldwide.org/educators/strategies-for-successful-learning/1096-deepening-mathematics-teaching-and-learning-through-the-concrete-pictorial-abstract-approach__

## Research to Support Using Technology to Create/View Visualizations

## Technology's Impact on Fraction Learning: An experimental comparison of virtual and physical manipulatives

This study attempted to maintain controlled conditions between two sets of classrooms. One set of classrooms used physical manipulatives to learn and demonstrate fraction concepts. The other set of classrooms used virtual manipulatives. The study found that students performed just as well, if not better when using the virtual manipulatives.Mendiburo, M. & Hasselbring, T. (2014). Technology's Impact on Fraction Learning: An experimental comparison of virtual and physical manipulatives. Journal of Computers in Mathematics and Science Teaching, 33(2), 209-231. Chesapeake, VA: Association for the Advancement of Computing in Education (AACE). Retrieved from http://www.editlib.org/p/48044/

## Virtual and Concrete Manipulatives: A Comparison of Approaches for Solving Mathematics Problems for Students with Autism Spectrum Disorder

In this study, small groups of students with autism were taught mathematics concepts using virtual and physical manipulatives. Students performed equally well using either of the types of manipulatives. However, the students reported that they enjoyed the virtual manipulatives more than physical manipulatives.Bouck, E. C., Satsangi, R., Doughty, T. T., & Courtney, W. T. (2014). Virtual and Concrete Manipulatives: A Comparison of Approaches for Solving Mathematics Problems for Students with Autism Spectrum Disorder. Journal of Autism and Developmental Disorders. 44(1), 180-193. Retreived from http://link.springer.com/journal/10803

## Visualizing Math: Computer Graphics Transforms the Face of Traditional Math Education

Lisa Stapleton’s article “Visualizing Math: Computer Graphics Transforms the Face of Traditional Math Education” discusses how the use of mathematical software helps students visualize math concepts as well as the success the students have. The article also talks about how much Computer-Aided Instruction (CAI)s are improving as well. Some of the software discussed are; Mathematica, a product produced by Wolfram Research Inc., Derive, by Soft Warehouse, f(x), authored by Martin Lapidus of Laxcaux Graphics, Fields & Operators among others.Stapleton, L. (June, 1990). Visualizing Math: Computer Graphics Transforms the Face of Traditional Math Education. Computer Graphics World, 58. Retrieved from http://www.cgw.com/Publications.aspx