**Research Based Purpose and Instruction**At Teach 4 Mastery we have looked and listened to what prominent researchers have shown works in math instruction. Our products align with the latest research and are designed, based on that research, to change the way you look at math. Perceptions Math is grounded in the use of the concrete representational abstract (CRA) instructional sequence. The use of CRA to teach struggling math learners has a strong research base (e.g. Butler, Miller, Crehan, Babbit, & Pierce, 2003; Maccini & Hughes, 2000; Maccini & Ruhl, 2000; Witzel, 2005). The program also includes the use of multiple strategy instruction, video anchored instruction, and reciprocal teaching, which are effective research-based practices (Gersten et al., 2009). Having observed different math curricula that used physical representations we were not satisfied that the manipulatives used, nor the methodologies taught, were as effective as they could be in helping students develop computational, conceptual, and application understanding. See the power of manipulatives like base-ten blocks and other similar rod style manipulatives we also observed what we considered to be weaknesses in the manipulatives abilities to move students forward in their math progression. Such as continuing to require students to count when moving into addition concepts, or taking large amounts of time assembling units or groups when time is a precious commodity not to be wasted. Thus we created the MasterPieces mastery math manipulatives. The manipulatives are unique and are capable of being used by kindergarten students as well as high school students to move from counting to algebraic concepts using the same manipulatives throughout their math journey. The MasterPieces transcend grade level instruction and provide years of concrete and representational examples for all arithmetic concepts taught in elementary and middle school math. To accompany the MasterPieces, we also created the MasterFractions fraction manipulatives to move students from whole numbers to understanding fractions, while maintaining a consistency with the conceptual basis of the MasterPieces.

### Instructional Methods:

Lesson Steps: Step 1: Both Teacher and Student Watch Video Step 2: Explicitly Instruct Student(s) Using Manipulatives Step 3: Student Proceeds to Workbook Pages for Practice Step 4: Student Demonstrates Mastery - Teaching Back to You To support the written curriculum and the physical manipulatives, we also provide lesson-by-lesson videos clearly demonstrating how to use the manipulatives to teach students about mathematics. Video lessons are first watched by the teacher and student(s) then the teacher works with the student to demonstrate the lesson contents using the manipulatives. There are three phases to the instructional process:### 1. Construct:

Demonstrate through both concrete and semi-concrete representations of mathematical concepts and ideas. Whenever the instructions indicate to construct, we recommend using the MasterPieces to build or draw the problem given and develop conceptual understanding.### 2. Express:

Demonstrate through artistic, written, and verbal expression, the fluency and accuracy of the concept and computation of the mathematical process. Whenever the instructions indicate to solve, show, draw, etc... we recommend drawing, written, and verbal communications to have students express what they have learned. We encourage teachers to get creative on how the students will express to them and/or to others the understanding they have of a given concept and process. Each lesson has key vocabulary that must be understanding must be demonstrated as part of the mastery process. Ideally, a student who has mastered a concept can hear the problem verbally and express back the computation, concept, and application of the problem in a written or verbal manner.### 3. Apply:

Teachers first demonstrate the correct application of the mathematical concept and computation in real world settings then have students model this in their own work.### Word Problems & Mastery Challenges

– Each Student Workbook lesson has multiple word problems designed to help the student demonstrate their understanding of the concepts learned. At the end of each lesson is a Mastery Challenge. This is where students are challenged to apply what they have learned into specific types of applications, typically multi-step word problems.### Reflection & Mastery

– What did you learn? Can you teach it to someone else? Students need to master skills at each level of arithmetic, including complex applications of those skills, in order to become a true master of their craft. Also, a student teaching back what has been learned is a critical demonstration of the student's ability to: • Grasp the problem • Choose a process • Develop a solution • Articulate the reasoning behind their choices • or… Reveal where there are gaps in their understanding### Additional Key Research the Affected the Development of Perceptions

Research^{1}has shown that understanding fractions and whole-number division are early predictors of a student’s mathematical achievement in high school. The research further supports that mastery of fractions and division is important to develop understanding of algebra and other high school mathematics. Watch a video of Dr. Seigler discussing the study and the implications. Research

^{2}has also shown that an intervention should include explicit instruction that is systematic, cumulative, and multi-sensory. Additionally, interventionists should be proficient in the use of visual representations of mathematical ideas. The

*Preceptions*Program uses

*MasterPieces*and

*MasterFractions*to support the instruction of place value, addition, subtraction, regrouping, multiplication, division, and fractions, and much more consistent with these research findings.

**References**

###### Butler, F. M., Miller, S. P., Crehan, K., Babbitt, B., & Pierce, T. (2003). Fraction instruction for students with mathematics disabilities: Comparing two teaching sequences. *Learning Disabilities Research & Practice*, *18*(2), 99-111.

###### Jayanthi, M., Gersten, R., & Baker, S. (2008). Mathematics instruction for students with learning disabilities or difficulty learning mathematics: A guide for teachers. *Center on Instruction*.

###### Maccini, P., & Hughes, C. A. (2000). Effects of a problem-solving strategy on the introductory algebra performance of secondary students with learning disabilities. *Learning Disabilities Research & Practice*, *15*(1), 10-21.

###### Maccini, P., & Ruhl, K. L. (2000). Effects of a graduated instructional sequence on the algebraic subtraction of integers by secondary students with learning disabilities. *Education and Treatment of Children*, 465-489.

###### Witzel, B. S. (2005). Using CRA to teach algebra to students with math difficulties in inclusive settings.*Learning Disabilities—A Contemporary Journal*, *3*(2), 49-60.

###### ^{1}A 20 year longitudinal study on Early Predictors of High School Mathematics Achievement by Robert S. Seigler et al.

^{2}Assisting students struggling with mathematics: Response to Intervention (RtI) for elementary and middle schools by Russell Gersten et al.

Case Study by Sarah Watt Ph.D. - Miami University on the Perceptions Blue Level 2016.
Case Study by Kaitlyn Brame - Webster University on Perceptions Blue Level 2016

**Concrete-Representational-Abstract:**This is a gradual instructional sequence that moves students fluidly between three phases of learning.

**C**= Students use concrete manipulatives to model and solve problems.

**R**= Students use pictorial representations to model and solve problems.

**A**= Students use abstract numbers and symbols to model and solve problems.

**Explicit Instruction:**The instructional approach includes clear modeling, thinking aloud during instruction, presentation of multiple examples, and immediate, corrective feedback.

**Verbalizing Thinking:**The tutor/teacher teaches the material to the student and then the student teaches the material back to them. During this guided practice opportunity teachers are able to hear the verbal thinking of students allowing for immediate feedback when necessary. This verbalization helps to anchor skills (Jayanthi, Gersten, & Baker, 2008).

**Multiple Strategy Use:**The student learns multiple ways to approach and solve a problems. In addition, to reinforcing traditional methods of multiplication, division, and fraction it also teaches alternate procedures using concrete, representational, and abstract examples.

**Progress Monitoring:**The materials provide opportunities for teachers to collect data daily as well as at the beginning and end of each unit.

**Self-Monitoring Academic Content:**At the end of each page within the Perceptions Math materials are red, yellow, and green lights. If students feel confident with the content and are ready to continue they color in the green light, if they need clarification or extra practice they color in the yellow light, and if they need to stop and go over the material a second time they color in red.