## Additional information

Dimensions | 8.5 × 11 in |
---|---|

Cover | Paperback |

Dimensions (W) | 8 1/2" |

Dimensions (H) | 11" |

Page Count | 156 |

Publisher | CRL |

Year Printed | 1993 |

Susan P. Miller

“Jack has to write three paragraphs. He has 30 minutes to write these paragraphs. How much time can he spend on each paragraph?”

If problems like this cause students’ heads to spin, stop the merry-go-round now with *Division Facts 0 to 81*. In a scaffolded series of lessons, students progress through three different levels of instruction: concrete, representational, and abstract. At the concrete level, they use objects (like pennies) to understand that a problem like 30 divided by 3 is the same as splitting 30 objects into three equal groups. At the representational level, they use pre-printed pictures to expand this understanding. 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. Timed practice activities, motivational “pig games,” and practical word problems round out the instruction at each level.

Dimensions | 8.5 × 11 in |
---|---|

Cover | Paperback |

Dimensions (W) | 8 1/2" |

Dimensions (H) | 11" |

Page Count | 156 |

Publisher | CRL |

Year Printed | 1993 |

**Study 1**

**Overview**

The purpose of this study was to show the effects of the concrete-to-semiconcrete-to-abstract instructional sequence with regard to teaching *Division Facts 0 to 81*. Three students who were at-risk for having learning disabilities participated. All were performing below grade level in math. Their ages ranged from 10 to 11 years. They were taught by their regularly assigned teacher who used the *Division Facts 0 to 81* program. The measure was the number of division 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 ten problems incorrectly and three problems correctly per minute during baseline. The cross-over effect occurred during semi-concrete instruction for all three students. (The cross-over effect occurs when a student starts solving more problems correctly than incorrectly.) For example, at the end of semi-concrete instruction, Student 1 solved eleven problems correctly and two problems incorrectly in one 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 14 division problems correctly and made no incorrect responses within one minute.

**Conclusions**

This study shows that the concrete-to-abstract instructional sequence can be effective in teaching students at-risk for LD to solve simple division problems. Their rate of solving problems increased as the students progressed through the instructional phases in the sequence. The cross-over effect occurred only after the instruction was 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**

A field test was conducted that involved 22 teachers and 109 elementary students who were experiencing difficulties learning math. This student group included 102 students with learning disabilities (LD), 5 students with emotional disabilities, and 2 students who were at-risk for school failure. The field test took place in seven school districts in both small-group (less than 7 students) and larger group (7 to 18 students) instructional arrangements. The teachers were trained to use programs in the *Strategic Math Series*. Different groups of students were taught addition facts, subtraction facts, multiplication facts, and division facts, depending on their needs.

**Results**

The 19 students who received instruction in the *Division Facts 0 to 81* program earned a mean score of 9% on the acquisition pretest and 81% on the posttest. With regard to fluency, the students solved an average of 8 problems per minute during the first abstract lesson and 15 problems per minute at the completion of the program.

**Conclusions**

The results show that students with learning difficulties in math are able to learn basic division facts through use of the Division Facts 0 to 81 program. The students acquire division knowledge and improve their ability to solve division facts with fluency.

**Reference**

Mercer, C. D., & Miller, S. P. (1992). Teaching students with learning problems in math to acquire, understand, and apply basic math facts. *Remedial and Special Education*, 13(3), 19-35, 61.

**Study 3**

**Overview**

Multiple field tests were conducted that involved 56 teachers and 248 elementary students who were experiencing difficulties learning math. These 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 correct answers

Figure 2: Number of digits correct per minute

The results for the *Division Facts 0 to 81* program are shown in the pair of bars at the far-right side of each figure. Students earned a mean score of 25% answers correct on the acquisition pretest and 96% on the acquisition posttest. They produced an average of 11 correct digits per minute in baseline and 22 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% correct level on acquisition tests in all areas.

**Reference**

Miller, S. P., & Mercer, C.D. (1998). *Strategic math series trainer’s guide*. Lawrence, KS: Edge Enterprises.

**Cecil D. Mercer, Ed.D.**

**Affliations**

- Distinguished Professor (Emeritus)

- Department of Special Education

- University of Florida

- Gainesville, FL

- Certified Learning Strategies Trainer

- University of Kansas Center for Research on Learning

- Lawrence, KS

**My Background and Interests**

I am retired as a Distinguished Professor of Special Education at the University of Florida. As a professor, I taught courses in learning disabilities, learning strategies, instructional methodology, and behavior management. My research interests focused on learning strategies and reading and mathematics interventions. I authored numerous books and curriculum materials, including *Students with Learning Disabilities*, *Teaching Students with Learning Problems*, *the Strategic Math Series*, and *Great Leaps Reading*. For 11 years, I was Co-Director of the University of Florida Multidisciplinary Diagnostic and Training Program. In addition, I served on the Board of Directors of the International Dyslexia Association and on the Professional Advisory Board of the Learning Disabilities Association of America. I was awarded the College of Education Teacher of the Year award three times at the University of Florida and also received the University of Florida Graduate School Advisor/Mentoring Award and the College of Education Lifetime Achievement Award. As an alumnus of the University of Virginia Curry School of Education, I was awarded the Outstanding Higher Education Faculty Award.

**The Story Behind the Strategic Math Series**

Many students with learning problems lack proficiency in basic number facts and are unable to acquire and maintain math facts at fluency levels. Given the problems that many students with learning problems exhibit with lower-level math skills and the importance of these skills to overall math achievement, the need existed for comprehensive programming to teach basic math facts. Susan Miller, one of my doctoral students at the University of Florida, also was interested in determining effective ways to teach mathematics. For her dissertation, 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. Because the CRA sequence is appropriate for teaching the understanding of math throughout the span of math concepts, skills, and word problems, we conducted 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 for enjoyable practice. Due to the research results, the success and excitement of students in the program, and the enthusiasm of teachers using the CRA teaching sequence, Don Deshler and Jean Schumaker encouraged and supported the creation of the

**My Thoughts about the Strategic Math Series**

The

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

Teachers report that the math strategies instruction is easy to implement and that the students really understand addition, subtraction, place value, multiplication, and division when they finish the instructional lessons. Teachers who use comprehensive math programs indicate that the

**My Contact Information**

Cecil D. Mercer

Email: cecilmercer@cox.net

Phone: 352-378-4849

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