1. Strategic Math Series Professional Developer’s Guide developed by Susan P. Miller and Cecil D. Mercer
The Learning Strategy Series
2006
The University of Kansas
Center for Research on Learning
Lawrence, Kansas
University of Kansas Center for Research on Learning 2002

2. Overview of the Series 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
Addition with Regrouping
Subtraction with Regrouping
University of Kansas Center for Research on Learning 2006
University of Kansas Center for Research on Learning 2002

3. Purpose of the Series
To provide students who are at risk for failure in school with:
the skills needed to solve basic math problems.
a foundation for solving more advanced problems in the future.
University of Kansas Center for Research on Learning 2006

4. Rationales for the Series
Students need:
manipulative experiences to promote conceptual understanding of basic math facts and place value.
conceptual understanding of math skills prior to rote memorization.
strategies for solving math problems at the abstract level.
University of Kansas Center for Research on Learning 2006

5. Rationales for the Series
Students need:
fluency-building practice to master and retain basic math facts and place value.
strategies for solving word problems.
successful experiences in math.
University of Kansas Center for Research on Learning 2006

6. Guiding Principles
The Concrete-Representational-Abstract (C-R-A) Teaching Method
Concrete: use objects (e.g., pennies, cubes, sticks)
Representational: use drawings (e.g., circles, boxes) and tallies
Abstract: use numbers (e.g., 2, 7)
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7. Guiding Principles Mastery Learning Facts learned to fluency
Fluency assures retention
Students then able to solve more complex problems
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8. Guiding Principles Use of Mnemonics
Enables students to engage in the problem-solving process
Enables students to apply this process to everyday situations
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9. Instructional Stages Stage 1 Pretest
Stage 2 Teach Concrete Application
Stage 3 Teach Representational Application
Stage 4 Introduce the “DRAW” or “FIND” Strategy
Stage 5 Teach Abstract Application
Stage 6 Posttest
Stage 7 Provide Practice to Fluency
University of Kansas Center for Research on Learning 2006

10. The “DRAW” Strategy (used to teach all facts)
D Discover the sign
R Read the problem
A Answer, or draw and check
W Write the answer
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11. The “FIND” Strategy (used to teach place value)
F Find the columns
I Insert the T
N Name the columns
D Determine the answer
University of Kansas Center for Research on Learning 2006

12. Instructional Procedures
Give an Advance Organizer
Describe and Model
Conduct Guided Practice
Conduct Independent Practice
Conduct Problem-Solving Practice
Administer Timed Probe
Administer Facts Review Probe
Conduct Pig Game Practice
Provide Feedback
University of Kansas Center for Research on Learning 2006

13. Solving Addition Problems
Concrete: manipulatives
2 _________
+ 3 _________
_________
Representational: pictures or tallies
3 _________
Abstract: words
6 balls
+ 1 ball
balls
3 birds sitting Sally has 4 books.
+ 5 birds sitting Irene has 3 books.
birds sitting They have books altogether.
University of Kansas Center for Research on Learning 2006

14. Solving Subtraction Problems
Concrete: manipulatives
7 _________
- 4 _________
_________
Representational: pictures or tallies
5 _________
_________
Abstract: words
7 apples baseball cards
- 5 apples given away
apples left in collection
Jennifer had 4 pens.
She lost 2 of them.
She has pens left.
University of Kansas Center for Research on Learning 2006

15. Solving Place Value Problems
Concrete
In 35 _________, there are ____ tens and ____ ones.
The number 24 is another name for _________ and ___ _______.
Representational
2 tens and 7 ones is another name for __________.
Abstract
3 tens and ones
+ 2 tens and ones
tens and ___ ones is another name for ____.
Jerry has 5 tens.
Linda has 3 tens.
They have ___ tens altogether.
Pop has 2 tens and 3 ones. Ray has 5 tens and 4 ones. How many tens and ones do they have altogether?
University of Kansas Center for Research on Learning 2006

16. Solving Multiplication Problems
Concrete: manipulatives
7 groups
of 3 ________
________
Representational: pictures or tallies
3 groups
of _________
_________
Abstract: words
3 plates My friend has 4 trains.
of 7 oatmeal cookies Each train has 6 cars.
oatmeal cookies There are cars in all.
Cindy has 2 tapes. Each tape has 4 songs on it.
How many songs are on the tapes?
University of Kansas Center for Research on Learning 2006

17. Solving Division Problems
Concrete: manipulatives
6 ___________ has how many groups of 2 _______?
Solve: __________ : __________ = ____________
Total Objects/group Groups
Representational: pictures or tallies
How many groups of 4 _______ are in 16 _______?
Total Objects/group Groups
Abstract: words
How many groups of 5 dogs are in a pack of 25 dogs?
Solve: __________ : __________ = ____________
University of Kansas Center for Research on Learning 2006

18. “FAST DRAW” (used to teach word problems)
F Find what you’re solving for
A Ask yourself, “What are the parts of the problem?”
S Set up the numbers
T Tie down the sign
D Discover the sign
R Read the problem
A Answer, or draw and check
W Write the answer
University of Kansas Center for Research on Learning 2006

19. Advanced Problem-Solving Practice
Addition:
Bob has 3 shirts. Tim has 6 shirts. How many shirts do they have altogether?
Subtraction:
Luis had 8 sheets of paper. Tom had 6 frogs. Luis gave his 8 sheets of paper to Sylvia. How many sheets of paper does Luis have left?
University of Kansas Center for Research on Learning 2006

20. Advanced Problem-Solving Practice
Place Value:
Henry planted 12 flowers. Hank planted 27 more flowers. Phillip planted 2 trees. How many flowers were planted in all? Answer: _____ This number has ___ tens and ___ ones.
Multiplication:
Jan has 8 dolls. Each doll has 2 hats. Jan also has 3 pets. How many hats are there in all?
University of Kansas Center for Research on Learning 2006

21. Advanced Problem-Solving Practice
Division:
Linda handed out 56 tickets. Ron wanted 10 tickets. Each student received 8 tickets. How many students received tickets?
Students also create their own word problems.
University of Kansas Center for Research on Learning 2006

22. Feedback Sequence
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 application.
6. Close the feedback session with positive statement.
University of Kansas Center for Research on Learning 2006

23. Field Test Results: Computation of Facts
mean mean
pretest posttest increase
Addition % 93% 59%
Subtraction % 96% 64%
Place Value 20% 90% 70%
Addition % 93% 44%
Subtraction % 95% 75%
Multiplication 48% 92% 44%
Division 25% 96% 71%
University of Kansas Center for Research on Learning 2006

24. Field Test Results: Rate Increases Following the Posttest Lesson
mean mean
beginning ending
rate* rate* increase
Addition %
Subtraction %
Place Value %
Addition %
Subtraction %
Multiplication %
Division %
Note: Most students in the field tests needed additional fluency practice beyond the 21st or 22nd lesson. Continuation of Pig Games, Math Minutes, and/or Facts Reviews is appropriate during concrete and representational lessons of subsequent skills.
* Digits correct per minute.
University of Kansas Center for Research on Learning 2006

25. Materials to Obtain Manipulative devices (3 types)
Chalkboard or overhead projector
Timer or clock with a second hand
All other materials contained in manuals.
Pig dice accompany each manual.
The color of the numbers on the dice match the color of the letters in the title of the manual.
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26. Student Notebook Contents
Contract
Progress Chart
Learning Sheets
Math Minute(s)
Facts Review(s)
Pig Game Answer Key
Relationship/Rules Sheet
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27. Addition 0-9 and the Concrete Process5
+ 2
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28. Addition 0-9 and the Representational Process3
+ 2 4
+ 3 5
+ 1
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29. Subtraction 0-9 and the Concrete Process5
- 3
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30. Subtraction 0-9 and the Representational Process5
- 2 4
- 3 6
- 4
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31. Place Value and the Concrete Process
Tens Ones
24 =
17 =
46 =
Tens Ones
Tens Ones
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32. Place Value and the Representational Process
14 =
Tens Ones
Tens Ones
Tens Ones
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33. Addition 10-18 and the Concrete Process3
+ 8
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34. Addition 10-18 and the Representational Process5
+ 6 6
+ 7 8
+ 5
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35. Subtraction 10-18 and the Concrete Process12
- 4
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36. Subtraction 10-18 and the Representational Process12
- 6 11
- 3 16
- 8
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37. Multiplication and the Concrete Process4
x 3
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38. Multiplication and the Representational Process4
x 2
3 x 5 = _____ 6
x 3
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39. Division and the Concrete Process
6 : 3 = ____
5 10
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40. Division and the Representational Process
8 : 2 = _____
12 : 6 = _____
6 : 2 = _____
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41. Subtraction Rules & Relationships
In subtraction, when the bottom number in the ones column is bigger than the top number in the ones column, the ten is traded.
To help you learn the subtraction trade rule, remember the BBB sentence: Bigger number on Bottom means Break down the ten and trade.
Example: 14 -7 ?
University of Kansas Center for Research on Learning 2006

42. Subtraction Rules & Relationships
Addition and subtraction facts have the same numbers, but in different orders.
e.g.: = = 7
Because addition and subtraction are related, you always can state a subtraction problem as an addition problem.
e.g.: = ? ? + 6 = 14
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43. Subtraction Rules & Relationships
To check a subtraction answer, add your answer to the number that is being subtracted. If the sum of these two numbers equals the first number in the subtraction problem, your answer is correct.
e.g.: = 9 Check: = 14
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44. Multiplication Rules and Relationships
Any number times 0 equals 0.
e.g.: 6 x 0 = 0 9 x 0 = 0
Any number times 1 equals the original number.
e.g.: 7 x 1 = 7 9 x 1 = 9
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45. Multiplication Rules and Relationships
2 times any number equals the number added to itself
e.g.: 2 x 8 = = 16
Changing the order of the numbers in multiplication does not change the answer.
e.g.: 6 x 3 = x 6 = 18
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46. Division Rules and Relationships
0 divided by any number equals 0.
0 ÷ 6 = 0 0 ÷ 4 = 0
Any number divided by 1 equals the number.
7 ÷ 1 = 7 2 ÷ 1 = 2
University of Kansas Center for Research on Learning 2006

47. Division Rules and Relationships
Any number divided by that number equals 1.
7 ÷ 7 = 1 2 ÷ 2 = 1
Multiplication and division facts have the same numbers, just in different orders.
8 x 7 = ÷ 7 = ÷ 8 = 7
University of Kansas Center for Research on Learning 2006

48. Division Rules and Relationships
Because multiplication & division are related, you can always state a division problem as a multiplication problem.
35 ÷ 7 = ? ? x 7 = 35 ?
To check a division answer, multiply your answer by the number that the total is being divided by. If the answer to this multiplication problem equals the total, your answer is correct.
35 ÷ 5 = 7 Check: 7 x 5 = 35
University of Kansas Center for Research on Learning 2006