1. “Please put away that pocket calculator, Billy.”

2. BASIC FACTS

3. I. Objectives for this Lesson Define basic facts for each of the operations, discuss their role and importance in a mathematics program, and identify reasonable outcomes for students. Define basic facts for each of the operations, discuss their role and importance in a mathematics program, and identify reasonable outcomes for students. Identify and discuss factors which contribute to a child’s success with basic facts. Identify and discuss factors which contribute to a child’s success with basic facts. Identify desired student outcomes with basic facts. Justify your decision. Identify desired student outcomes with basic facts. Justify your decision. Define Brownell’s stages for the successful learning of the basic facts and elaborate on examples and techniques at each level. List techniques Brownell identified as being inappropriate in learning the basic facts. What happens if the children begin the drill stage too soon? Define Brownell’s stages for the successful learning of the basic facts and elaborate on examples and techniques at each level. List techniques Brownell identified as being inappropriate in learning the basic facts. What happens if the children begin the drill stage too soon?

4. II. What Is a Basic Fact? 3 + 4 7 2 + 7 9 2 + 9 11 9 + 9 18 0 + 0 0 7 +11 18 3 +10 13 20 +30 50 12 + 0 12 10 + 1 11

5. 3 x 4 12 5 x 6 30 0 x 1 0 9 x 9 81 1 x 8 8 10 x 4 40 12 x 6 72 11 x 11 121 12 x 0 0 10 x 10 100

6. III. What Is Basic About Basic Facts?

7. IV. Why Does a Child Need to Know Basic Facts?

8. V. What Does a Child Need to Know About Basic Facts?

9. VI. What is Known About Learning Basic Facts? Understanding first - drill later Understanding first - drill later Experiences with concrete materials Experiences with concrete materials There are specific stages of development There are specific stages of development A careful instructional sequence yields success A careful instructional sequence yields success Use effective drill techniques after earlier stages are developed Use effective drill techniques after earlier stages are developed Attitudes are critical! Attitudes are critical!

10. Brownell’s Stages of Learning the Basic Facts Effective Techniques 1.Naive counting 2.Sophisticated counting 3.Strategies (solution techniques) 4.Immediate recall (meaningful habituation) Ineffective Techniques 1. Guessing 2. Recitation of tables 3. Rote recall without understanding understanding

11. Comparison of Stages Stage1.2.3.4.AdditionRecountingCounting-OnStrategiesDrill Multiplication Counting Skip Counting Strategies Drill

12. VII. Do All Children Go Through All of the Stages?

13. VIII. The Role of Drill in Learning Basic Facts

16. IV. Uses of Number Cardinal (Discrete) - How many? Cardinal (Discrete) - How many? Cardinal (Continuous) - How much? Cardinal (Continuous) - How much? Ordinal - Which one? Ordinal - Which one? Nominal - A name or code Nominal - A name or code

17. Counting-on Four and three more makes: four, five, six, seven Four and three more makes: four, five, six, seven This is a critical counting skill! This is a critical counting skill! Activities Activities

18. Cover Up

19. Counting-back Rote counting backwards Rote counting backwards Why is counting-back important? Why is counting-back important?

20. Invariance of Number The most critical “hurdle” of the primary grades is conservation (invariance) of number. The most critical “hurdle” of the primary grades is conservation (invariance) of number. Can we teach invariance of number? Can we teach invariance of number? Can we create an environment where invariance is learned? Can we create an environment where invariance is learned? Activities Activities

21. Skip Counting Two, four, six, eight, ten,... Two, four, six, eight, ten,... A prerequisite to place-value, basic facts, multiplication, division, time, money,... A prerequisite to place-value, basic facts, multiplication, division, time, money,... Seldom mentioned directly in textbooks other than as sequences to be completed Seldom mentioned directly in textbooks other than as sequences to be completed 2 4 6 __ 10 Activities Activities

22. Extending to Twenty The child’s first work beyond ten should not emphasize place-value. Rather, it should be an extension of the familiar work with numbers of ten or less. One ten vs. ten ones is very confusing. The child’s first work beyond ten should not emphasize place-value. Rather, it should be an extension of the familiar work with numbers of ten or less. One ten vs. ten ones is very confusing. There are significant problems with the number names between ten and twenty. There are significant problems with the number names between ten and twenty. Activities Activities

23. Making the Connections Models Concrete/Iconic Language Oral/Written Symbols

24. III. Operation “Situations ” Almost all uses of the four basic operations may be organized into just eight “situations” Almost all uses of the four basic operations may be organized into just eight “situations” There is no reason to assume that instruction in one “situation” will generalize into another “situation”. Instruction must be provided in each. There is no reason to assume that instruction in one “situation” will generalize into another “situation”. Instruction must be provided in each.

26. Operation “Situations” Addition and Subtraction Join Problems Join Problems Separate Problems Separate Problems Part-Part-Whole Problems Part-Part-Whole Problems Compare Problems Compare Problems

28. IV. Stages in Developing the Operations

29. Operation “Situations” Multiplication and Division Equal Groups (partition and measurement) Equal Groups (partition and measurement) Multiplicative Comparison Multiplicative Comparison Combinations Combinations Product of Measures Product of Measures

30. Activity: Vertical Addition Sentence Bowls Adapted from “Math Their Way” by Mary Baratta-Lorton, Addison- Wesley, 1995

31. 66 2 4 2 4 + 6 2+ 4 6 Make a Record

32. Activity: Horizontal Addition Sentence Helping Hands Adapted from “Math Their Way” by Mary Baratta-Lorton, Addison- Wesley, 1995

33. 5 53232 5 = 5 32+ + 3+2 Make a Record

34. Activity: Horizontal Subtraction Sentence Take-away hand Adapted from “Math Their Way” by Mary Baratta-Lorton, Addison- Wesley, 1995

36. Activity: Multiplication Cups-n-Counters

37. Activity: Division (Cars & Boats)

38. Instructional Sequence and Activities BASIC FACTS

39. I. Objectives for this Lesson Identify appropriate instructional techniques for each of Brownell’s stages. Identify appropriate instructional techniques for each of Brownell’s stages. Identify inappropriate instructional techniques for helping children learn the basic facts. Identify inappropriate instructional techniques for helping children learn the basic facts. Identify a wide range of strategies for each of the operations. Give examples of instructional activities for each technique. Identify a wide range of strategies for each of the operations. Give examples of instructional activities for each technique. Identify an overall plan for teaching the basic facts. What does it mean to teach “strategy selection”? Identify an overall plan for teaching the basic facts. What does it mean to teach “strategy selection”?

40. II. An Instructional Sequence

41. Ages and Stages

42. A. Operation concepts and B.Naive counting

43. Activity: Vertical Addition Sentence Bowls Bowls Adapted from “Math Their Way” by Mary Baratta-Lorton, Addison- Wesley, 1995

44. Activity: Horizontal Addition Sentence Helping Hands Adapted from “Math Their Way” by Mary Baratta-Lorton, Addison- Wesley, 1995

45. Activity: Horizontal Subtraction Sentence Take-away hand Take-away hand Adapted from “Math Their Way” by Mary Baratta-Lorton, Addison- Wesley, 1995

46. Activity: Multiplication Cups-n-Counters Cups-n-Counters

47. Activity: Division (Cars & Boats)

48. C. Sophisticated Counting Use counting activities Use counting activities “Generate” the Basic Facts “Generate” the Basic Facts Begin to use tables to record solutions Begin to use tables to record solutions Consider the multiple situations for each operation Consider the multiple situations for each operation

50. D. Strategies Introduce the strategy Introduce the strategy Practice the strategy Practice the strategy Add new strategies Add new strategies Individualize Individualize Practice strategy selection Practice strategy selection

51. E. Drill Make sure students are successful in previous stages first. Make sure students are successful in previous stages first. Use small drills, distributed over time. Use small drills, distributed over time. Provide immediate feedback at first. Provide immediate feedback at first. Use “disguised practice” to maintain motivation Use “disguised practice” to maintain motivation Continue to focus on success and positive attitudes. Continue to focus on success and positive attitudes.

52. III. Other Issues Fingers vs. concrete materials Fingers vs. concrete materials Upper grade remediation Upper grade remediation “Help” from parents “Help” from parents