## Additional information

Dimensions | 8.5 × 11 in |
---|---|

Cover | Paperback |

Dimensions (W) | 8 1/2" |

Dimensions (H) | 11" |

Page Count | 124 |

Publisher | CRL |

Year Printed | 1994 |

Cecil D. Mercer

Does adding the bigger numbers, like 8 + 9, make students to break into a cold sweat or send them scurrying to the nurse’s office? If so, provide the skills they need to tackle those math facts; teach them how to add numbers that total up to18.

Using the concrete-representational-abstract method of learning, students learn to really understand what problems like 7 + 8 and 9 + 6 mean. At the concrete level, students count out objects like checkers or sticks to understand that the problem 7 + 8 really means 7 objects being added to 8 objects (or vice versa). Once they have acquired this level of understanding, they move on to the representational level of learning. Here, they create pictures of boxes or circles next to each of the numbers. Finally, they graduate to the abstract level, where they use numbers alone to answer the problem, or, if they’re unable to recall the answer, draw out the problem using tallies.

Along the way, students also become proficient solvers of word problems involving addition facts 10 to 18. They also have fun practicing these facts with the colorful “pig dice” that come with the book, and they learn to increase the speed at which they are able to recall these foundational facts.

Dimensions | 8.5 × 11 in |
---|---|

Cover | Paperback |

Dimensions (W) | 8 1/2" |

Dimensions (H) | 11" |

Page Count | 124 |

Publisher | CRL |

Year Printed | 1994 |

**Study 1**

**Overview**

The purpose of this study was to show the effects of the concrete-to-semiconcrete-to-abstract instructional sequence that is used in all programs in the *Strategic Math Series*. More specifically, this study showed the effects of the instructional sequence with regard to teaching *Addition Facts 10 to 18*. Three students with learning disabilities participated. All were performing below grade level in math. Their ages ranged from 7 to 9 years. They were taught by their regularly assigned teacher who used the *Addition Facts 10 to 18* program. The measures were the number of addition problems that students solved correctly and incorrectly in a minute. A multiple-baseline across-students design was utilized.

**Results**

During baseline, all three students solved more problems incorrectly than correctly within one minute. For example, Student 1 solved about three problems incorrectly and zero problems correctly per minute during baseline. The cross-over effect occurred during concrete instruction for two students and during semi-concrete instruction for the third student. (The cross-over effect occurs when a student starts solving more problems correctly than incorrectly.) For example, at the end of concrete instruction, Student 1 solved three problems correctly and one problem incorrectly per minute. Once the cross-over effect had occurred, the students’ rates for correct responses continued to increase, and their rate of incorrect responses decreased or remained very low. At the end of the abstract instruction, Student 1, for example, solved 8 addition problems correctly and had no incorrect responses per minute.

**Conclusions**

This study shows that the concrete to abstract instructional sequence can be effective in teaching students with LD to solve addition problems. Their rate of solving problems increases as the students progress through the instructional phases in the sequence. The cross-over effect occurs only after the instruction is instituted as shown by the multiple-baseline design.

**Reference**

Miller, S. P., & Miller, C. M. (1993). Using data to learn about concrete-semiconcrete-abstract instruction for student with math disabilities. *Learning Disabilities Research & Practice*, 8(2), 89-96.

**Study 2**

**Overview**

Multiple field tests were conducted that involved 56 teachers and 248 elementary students who were experiencing difficulties learning math. The field tests took place in seven school districts in self-contained, resource, and general education classes. The teachers were trained to use programs in the Strategic Math Series. Different groups of students were taught addition facts, subtraction facts, multiplication facts, division facts, and place value concepts and skills, depending on their needs.

**Results**

Substantial gains were made by the students in all areas. See the figures below for the results in each math area. Figure 1 shows the results on untimed acquisition tests, and Figure 2 shows the results on timed proficiency tests (i.e., fluency tests). The number of students participating in each field test is shown beneath each pair of bars on the graph.

Figure 1: Percentage of problems solved correctly

Figure 2: Number of digits correct per minute

The results for the *Addition Facts 10 to 18* program are shown in the fourth pair of bar graphs in each figure. Students earned a mean score of 49% correct answers on the acquisition pretest and 93% on the posttest. They had an average of 14 correct digits per minute in baseline and 24 correct digits per minute after instruction.

**Conclusions**

The programs in the *Strategic Math Series* produce significant gains in student performance on math acquisition and fluency tests across several areas of mathematics. In addition, these programs all produce socially significant final performances with students earning scores around or above the 90% level on acquisition tests in all areas.

**Reference**

Miller, S. P., & Mercer, C.D. (1998). *Strategic Math Series professional developer’s guide*. Lawrence, KS: Edge Enterprises.

**Susan P. Miller, Ph.D.**

**Affliations**

- Professor

- Department of Special Education

- University of Nevada Las Vegas

- Las Vegas, NV

- Certified SIM Professional Development Specialist

- University of Kansas Center for Research on Learning

- Lawrence, KS

**My Background and Interests**

I am a Professor of Special Education at the University of Nevada Las Vegas (UNLV). In this role, I teach courses in learning strategies, instructional methodology, and leadership. My research interests focus on learning strategies and mathematics interventions. I’ve had the opportunity to share much of what I know as author of *Validated Practices for Teaching Students with Diverse Needs and Abilities and as co-author of Designing and Implementing Mathematics Instruction for Students with Diverse Learning Needs*, and the *Strategic Math Series*. Prior to joining the faculty at UNLV, I worked as Program Administrator for the Multidisciplinary Diagnostic and Training Program at the University of Florida. Additionally, I’ve taught students with and without disabilities at the elementary and secondary levels. As a high-school general education teacher, I taught social science courses and compensatory mathematics. As a junior-high general education teacher, I taught geography and American history. As a middle-school special education teacher, I taught reading, and as an elementary diagnostic classroom teacher, I taught math, reading, and language arts to students with medical, learning, and behavioral challenges.

**The Story Behind the Strategic Math Series**

Arithmetic and mathematics were my least preferred subjects in the school curricula as a child and teenager. I spent a good bit of time memorizing procedures to get correct answers without truly understanding the meaning behind those procedures. From an early age, becoming a teacher was high on my list of goals, but I never imagined that mathematics would be the subject I’d teach. As fate would have it, my teaching assignments at both the elementary and high school levels included mathematics. It wasn’t until I began teaching math that I realized it was not math that I had disliked all those years, but it was the way I had been taught math. That realization launched my dedication to finding better ways to teach this complex area of the curriculum. Cecil Mercer, my doctoral mentor at the University of Florida (Go Gators!), also was interested in determining effective ways to teach mathematics. Thus, when dissertation time arrived, we designed a study that involved the use of the Concrete-Representational-Abstract (CRA) teaching sequence to help students acquire an understanding of place value. The positive results obtained in this dissertation study caused us to launch a series of studies and field tests related to teaching basic math facts using the CRA sequence with integrated strategy instruction, a graduated word problem sequence, math timings, and numerous PIG dice games to make math practice fun. The positive outcomes for students and positive feedback from their teachers motivated us to share our results with Jean Schumaker and Don Deshler, and shortly thereafter the

**My Thoughts about the Strategic Math Series**

I have enjoyed witnessing the positive effects of

**Teacher and Student Feedback on the Strategic Math Series**

Teachers routinely tell me that the math strategies instruction is easy to implement and that their students love it. They report that the students really understand addition, subtraction, place value, multiplication, and division when they finish the instructional lessons. Teachers who use a variety of comprehensive mathematics programs say that the Strategic Math Series is a wonderful supplement to these programs. They note that students need this supplemental instruction and practice to be successful in math. When Cecil Mercer and I conducted the field tests for the series, we had opportunities to talk with many teachers and to read their written feedback about the lessons. The 56 teachers involved in these field tests were overwhelmingly positive and indicated they would continue to use the program even though the field tests were complete. Written feedback from the students was very positive as well. I recently received a letter from a parent-volunteer who was using the program with her son and another child in an elementary resource room setting. She noted that the students were engaged and enthusiastic and wrote, “I’m so excited about how rapidly both children are progressing and what good feelings the children are experiencing due to their success.”

**My Contact Information**

Susan P. Miller, Ph.D.

Professor

Department of Special Education-Box 453014

University of Nevada Las Vegas

Las Vegas, NV 89154

Email: millersp@unlv.nevada.edu

Work Phone: 702 895-1108

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