1. Addition With Regrouping and Subtraction With Regrouping Trainer’s Guide developed by Bradley J. Kaffar and Susan P. Miller
Strategic Math Series: Level 2
2011
The University of Kansas
Center for Research on Learning
Lawrence, Kansas
University of Kansas Center for Research on Learning 2002

2. Overview of the Strategic Math Series
Level 1
Addition Facts 0-9
Subtraction Facts 0-9
Place Value: Discovering Ones & Tens
Addition Facts 10-18
Subtraction Facts 10-18
Multiplication Facts 0-81
Division Facts 0-81
Level 2
Addition With Regrouping
Subtraction With Regrouping
University of Kansas Center for Research on Learning 2011
University of Kansas Center for Research on Learning 2002

3. Purpose of the Series (Level 2) To provide students with:
the skills needed to solve complex computation and word problems.
a foundation for solving more advanced problems in the future.
University of Kansas Center for Research on Learning 2011

4. Rationales for the Series
Students need:
manipulative and pictorial models to promote conceptual understanding of algorithms and place value.
strategies for solving math problems at the abstract level.
skills to discriminate between problems that require regrouping and problems that do not require regrouping.
University of Kansas Center for Research on Learning 2011

5. Rationales for the Series
Students need:
fluency-building practice to master and retain complex computation and regrouping skills.
strategies for solving word problems.
successful experiences in math.
University of Kansas Center for Research on Learning 2011

6. Guiding Principles
The Concrete-Representational-Abstract (C-R-A) Teaching Method
Concrete: use objects (e.g., base-ten blocks)
Representational: use drawings (e.g., squares, lines) and tallies
Abstract: use numbers (e.g., 42, 317)
University of Kansas Center for Research on Learning 2011

7. Guiding Principles Mastery Learning Procedural processes become fluent
Fluency assists retention
Students then ready to advance to more complex problems
University of Kansas Center for Research on Learning 2011

8. Guiding Principles Use of Mnemonics
Enables students to actively engage in the problem-solving process
Enables students to apply this process independently
University of Kansas Center for Research on Learning 2011

9. Instructional Phases Phase 1 Pretest
Phase 2 Teach Concrete Application
Phase 3 Teach Representational Application
Phase 4 Introduce the “RENAME” Strategy
Phase 5 Teach Abstract Application
Phase 6 Posttest
Phase 7 Practice to Fluency
University of Kansas Center for Research on Learning 2011

10. Concrete Level Lessons 1 - 3
Use three-dimensional objects (e.g., base-ten blocks) and a place-value mat to represent and solve computation problems.
Ones blocks
Tens strips
Hundreds flats
University of Kansas Center for Research on Learning 2011

11. Representational Level
Lessons 4 and 5
Use drawings to solve computation problems.
Horizontal tallies represent ones
Vertical lines represent tens
Squares represent hundreds
University of Kansas Center for Research on Learning 2011

12. Representational Level Addition With Regrouping
University of Kansas Center for Research on Learning 2011

13. Representational Level Subtraction With Regrouping
University of Kansas Center for Research on Learning 2011

14. E Examine the ones column: 10 or more, go next door.
The “RENAME” Strategy (used to solve addition with regrouping problems)
R Read the problem.
E Examine the ones column: 10 or more, go next door.
N Note the ones in the ones column.
A Address the tens column: 10 or more, go next door.
M Mark tens in the tens column.
E Examine and note hundreds; exit with a quick check.
University of Kansas Center for Research on Learning 2011

15. “10 or More” Sentences
Adding the Ones: If adding the numbers in the ones column results in 10 or more, regroup to form a ten (10 or more, go next door). Adding the Tens: If adding the numbers in the tens column results in 10 or more, regroup to form a hundred (10 or more, go next door).
University of Kansas Center for Research on Learning 2011

16. Addition With Regouping
West Region Update Conference for SIM Professional Developers
Read the problem. 3 5 7 + 2 8 4
University of Kansas Center for Research on Learning 2011
Copyright © 2009 by Susan P. Miller and Bradley J. Kaffar

17. Examine the ones column: 10 or more, go next door.3 5 7 + 2 8 4
If adding the numbers in the ones column results in 10 or more, regroup to form a ten (10 or more, go next door).
University of Kansas Center for Research on Learning 2011

18. Note ones in the ones column.1 3 5 7 + 2 8 4
University of Kansas Center for Research on Learning 2011

19. Address the tens column: 10 or more, go next door.3 5 7 + 2 8 4
If adding the numbers in the tens column results in 10 or more, regroup to form a hundred (10 or more, go next door).
University of Kansas Center for Research on Learning 2011

20. Mark tens in the tens column.1 3 5 7 + 2 8 4
University of Kansas Center for Research on Learning 2011

21. Examine and note hundreds; exit with a quick check.
Addition With Regouping
West Region Update Conference for SIM Professional Developers
Examine and note hundreds; exit with a quick check. 1 3 5 7 + 2 8 4 6
University of Kansas Center for Research on Learning 2011
Copyright © 2009 by Susan P. Miller and Bradley J. Kaffar

22. E Examine the ones column: use the BBB Sentence for ones.
The “RENAME” Strategy (used to solve subtraction with regrouping problems)
R Read the problem.
E Examine the ones column: use the BBB Sentence for ones.
N Note the ones in the ones column.
A Address the tens column: use the BBB Sentence for tens.
M Mark tens in the tens column.
E Examine and note hundreds; exit with a quick check. 22
University of Kansas Center for Research on Learning 2011

23. BBB Sentences
BBB Sentence for Ones: Bigger number on Bottom means Break down a ten and trade. BBB Sentence for Tens: Bigger number on Bottom means Break down a hundred and trade.
University of Kansas Center for Research on Learning 2011

24. Subtraction With Regrouping
West Region Update Conference for SIM Professional Developers
Read the problem. 4 5 7 - 2 6 8
University of Kansas Center for Research on Learning 2011
Copyright © 2009 by Susan P. Miller and Bradley J. Kaffar

25. Examine the ones column: use the BBB Sentence for ones.4 17 5 7 - 2 6 8
Bigger number on Bottom means Break down a ten and trade.
University of Kansas Center for Research on Learning 2011

26. Note ones in the ones column.4 17 5 7 - 2 6 8 9
University of Kansas Center for Research on Learning 2011

27. Address the tens column: use the BBB Sentence for tens.3 14 17 4 5 7 - 2 6 8 9
Bigger number on Bottom means Break down a hundred and trade.
University of Kansas Center for Research on Learning 2011

28. Mark tens in the tens column.3 14 17 4 5 7 - 2 6 8 9
University of Kansas Center for Research on Learning 2011

29. Examine and note hundreds; exit with a quick check.
Subtraction With Regrouping
West Region Update Conference for SIM Professional Developers
Examine and note hundreds; exit with a quick check. 3 14 17 4 5 7 - 2 6 8 1 9 1 8 9 + 2 6 4 5 7
University of Kansas Center for Research on Learning 2011
Copyright © 2009 by Susan P. Miller and Bradley J. Kaffar

30. Instructional Procedures
Give an Advance Organizer
Describe and Model
Conduct Guided Practice
Conduct Independent Practice
Conduct Problem-Solving Practice
Administer Regrouping Probe
Administer Review Probe
Conduct Pig Game Practice
Provide Feedback
University of Kansas Center for Research on Learning 2011

31. Feedback Sequence for Individuals
1. Score each student’s product for correct and incorrect responses.
2. Meet with student; plot score on the Progress Chart.
3. Specify incorrect responses and error patterns if they exist.
4. Demonstrate the task.
5. Ask student to practice the application.
6. Close the feedback session with positive statement.
University of Kansas Center for Research on Learning 2011

32. Feedback Sequence for Groups
1. Score the student products.
2. Provide positive feedback to the group.
3. Specify incorrect responses and corresponding error patterns if they exist.
4. Show the students how to avoid the error pattern.
5. Ask the students to practice the application.
6. Close the feedback session with positive statement.
University of Kansas Center for Research on Learning 2011

33. Field Test Results Addition With Regrouping: Study 1
Computation
Mean Pretest
Mean Posttest
Treatment Group
49%
90%
Comparison Group
66%
72%
Word Problems
41%
89%
51%
69%
Addition With Regrouping Minute*
3.22
8.11
6.00
*Problems correct per minute
University of Kansas Center for Research on Learning 2011

34. Field Test Results Addition With Regrouping: Study 2
Mean
Baseline
Treatment
Maintenance
Generalization
Triad 1
23.33%
92.74%
93.33%
90.00%
Triad 2
22.50%
94.43%
96.67%
Triad 3
5.00%
91.29%
Participants were nine elementary students with learning disabilities.
All nine participants increased their performance level only after introduction of treatment.
Maintenance occurred seven days after instruction ceased.
Generalization occurred 14 days after instruction in the general education classroom.
Percentage of non-overlapping data (PND) was 97.83% (i.e., a large effect size).
University of Kansas Center for Research on Learning 2011

35. Field Test Results Subtraction With Regrouping: Study 1
Computation
Mean Pretest
Mean Posttest
Treatment Group
60%
86%
Comparison Group
76%
78%
Word Problems
45%
77%
62%
57%
University of Kansas Center for Research on Learning 2011

36. Field Test Results Subtraction With Regrouping: Study 2
Mean Baseline
Mean Treatment
Mean Maintenance
10.95%
87.20%
86.00%
Participants were five fifth grade students with learning disabilities.
All five participants increased their performance level only after introduction of treatment.
Maintenance occurred seven days after instruction ceased.
Percentage of non-overlapping data (PNP) was 90.20% (i.e., a large effect size).
University of Kansas Center for Research on Learning 2011

37. Materials to Obtain Base-ten blocks Whiteboard and display equipment
Stopwatch or clock with a second hand
All other materials are contained in manuals.
Pig dice accompany each manual.
University of Kansas Center for Research on Learning 2011

38. Student Notebook Contents
Learning Contract
Progress Chart
Place-Value Mats: Hundreds, Tens, and Ones
Learning Sheets
Regrouping Minute(s)
Review Minute(s)
Pig Game Sheets
Celebrate your accomplishments!
University of Kansas Center for Research on Learning 2011