1. Agenda  Review application exercises and hand in  Error correction & diagnosis  Math vocabulary  Counting  Symbol Identification & Place Value  Curriculum Evaluation Assignment

2. TEACHING VOCABULARY AND LANGUAGE SKILLS

3. Two Areas:  Language of instruction  Mathematics-related vocabulary and language skills

4. Language of Instruction  Terms commonly used in directions given by teachers (directions, actions, names of objects, names of colors).  Students should be screened to ensure they possess the language concepts and if not they should receive remediation.  Remediation:  Place in math program with carefully controlled teacher wording and provide supplementary language instruction

5. Math-related Vocabulary and Language Skills  Terms used to describe characteristics of objects  e.g., square, circle, dime,  Terms used to describe relationships between objects  e.g., parallel, similar, near, far

6. Math-related Vocabulary and Language Skills  Terms used to describe numbers in an operation and the operations themselves  e.g., sum, addend, difference, add, subtract  Classification terms  e.g., 6 boys, 7 girls, 3 cats

7. Guidelines  Need to integrate brief vocabulary-oriented instructional activities into math curriculum  Sequence of instruction depends on necessity of term. Some terms must be taught as preskills, others can wait until strategy is taught.  Preskill -- end with, side, equal, same, other  Later -- denominator, numerator, subtrahend

8. Vocabulary Teaching Procedures  Modeling positive and negative examples  Using synonyms  Giving definitions

9. Modeling Positive and Negative Examples  Model positive and negative examples of the new word  Test the students on their mastery of the examples  Present examples of the new word along with examples of other previously taught words

10. Positive/Negative Examples This is red This is not red This is red This is not red This is red This is not red

11. Presentations:  Quickly paced  Stress important words (this is not)  Present until all students are able to respond correctly to a group of three positive and three negative examples

12. Teaching Vocabulary with Synonyms  Teacher links new word with previously learned words rather than modeling examples  Must carefully select word used as a synonym -- be sure word is familiar  Tests with positive and negative examples  Provide practice in applying several recently taught synonyms

13. Format  Model and immediate acquisition  “Here is a new word. Subtract. Subtract means minus. What does subtract mean?”  Positive and negative examples  Write 4 + 2 on the board. “Do we subtract in this problem?”  Write 6-3 on the board. “Do we subtract in this problem?”  Review in context of other words.  What does ADD tell us to do? (plus)  What does SUBTRACT tell us to do? (minus)

14. Teaching Vocabulary with Definitions  Teach definition  Must carefully select words used in definition -- be sure word is familiar (i.e., a preskill).  Show positive and negative examples  Contrast it with previously learned definitions

15. Format  Model and immediate acquisition  A sum is the answer when you add. What is the sum?  Positive and negative examples  Write 4-1=3. Ask, “Is 3 a sum? How do you know?”  Write 4+2=6. Ask “Is 6 a sum? How do you know?”  Review in context of other words  What is the DIFFERENCE of 5 and 2?  What is the SUM of 5 and 2?

16. Critical Preskills  Equality  More-Less

17. Equality  Teach first in a context other than addition  Teach functional definition  Present series of positive and negative examples

18. More-Less  Important in story problems  Introduce as synonym (bigger, not bigger)  Present series of positive and negative examples

19. COUNTING

20. Instructional Analysis Questions to ask yourself for each type of counting:  What are the preskills?  What is this a preskill for?  What sequencing guidelines apply?  What are potential errors?  How do I correct them (remediation)?

21. Preskills  What are preskills?  Give an example of a skill that is a preskill for counting to ten.

22. Sequence & Integration General Guidelines  Preskills are taught before they are needed in strategies.  Easy skills are taught before more difficult ones.  Strategies and information that is likely to be confused are spaced or separated.

23. Types of math knowledge errors  Fact  Component  Strategy  Incorrect operation  Random errors

24. Counting  Why is counting important?  What is rote counting?  How is it different from rational counting? (What is the preskill for rational counting? Which sequencing guideline?) (Rational counting of 2 groups is a preskill for what? Which sequencing guideline?)

25. Counting  What is counting from a number? (What is counting from a number a preskill for? Which sequencing guideline is this?)

26. Counting  What is skip counting?  Why should skip counting by 10 be taught early?  What other skill does skip counting facilitate?  Which of the sequencing guidelines do these exemplify?

27. Rote Counting  How do you determine where to start rote counting with young children?  How do you teach rote counting? (See Summary Box 4.1 and Format 4.1)

28. Rote Counting: Error Correction How do you correct students who leave out a number when rote counting?

29. Correction Procedures  “Stop”  Model, lead, test the “hard part” (2 numbers prior to the error)  Test the whole sequence  Delayed test

30. Rote Counting: Practice and Review How can a teacher provide enough practice in order for lower performing students to master rote count?

31. Rational Counting Again, what is it? Why start with pictures rather than manipulatives? Format 4.2—How is rational counting taught?

32. Rational Counting: Error Correction What 2 types of errors can students make?

33. Rational Counting: Error Correction How do you correct coordination errors? How do you correct rote counting errors?

34. Rational Counting: Error Correction How do you correct coordination errors? 1. Tell the students to count only when they touch (you can model too). 2. “Test”—repeat the exercise. 3. Continue until students can count correctly several (3) times. 4. Delayed “test”—repeat the exercise later. (Provide lots of practice and review.)

35. Rational Counting: Error Correction How do you correct rote counting errors? 1. Model the hard part. 2. Lead students on the hard part. 3. “Test”—repeat the exercise (from 1). 4. Continue until students can count correctly several (3) times. 5. Delay “test”—repeat the exercise later. (Provide lots of practice and review.)

36. Rational Counting: Two Groups Why? What error might students make? How do you correct?

37. Counting from Different Numbers Why? How? What error might the students make? How do you correct this error?

38. Counting Backwards Why? How?

39. Rote Counting by 1s from 30 to 100  Preskills: Rote counting from a number other than 1; skip counting by 10s  Important skill to practice is counting across "decades."  Demonstrate the relationship between tens groupings (i.e., sequence of numerals 1, 2, 3...21, 22, 23).

40. Instructional Sequence  Count numbers higher than 100, stay within centuries and decades,  Count numbers higher than 100, stay within centuries, but count across decades,  Count across centuries beginning and ending at number ending with 5  After mastery, change examples to promote generalization.

41. Skip Counting: Count-by Series Why?  Why should counting by 10 be taught early?  What other skill do count by series facilitate?  Which of the sequencing guidelines do these exemplify?

42. Skip Counting: Count-by Series Why is it suggested we put count-by series in the following order (sequencing guideline): 10, 2, 5, 9, 4, 25, 3, 8, 7, 6

43. Skip Counting: Count-by Series The format (4.5) has 2 parts. What are they for? How do you teach a new series? When can the next series be started?

44. SYMBOL IDENTIFICATION AND PLACE VALUE

45. Symbol Identification and Place Value  Three major areas:  reading and writing numerals  column alignment  expanded notation

46. Terms  What do the following terms mean:  Number  Numeral  Place value  Expanded notation  Column alignment

47. Introducing the Concept  Concepts for kindergarten through early 1st grade  Numeral identification (0-10),  Numeral writing (0-10),  Symbol identification (+, -, =,  ),  Equation reading and writing,  Numeral and line matching.

48. Introducing Numeral Identification  When do you start?  What sequencing guideline is critical in determining the order in which numerals are introduced?

49. Introducing Numeral Identification  Order of introduction: what numerals would you separate?  Rate of instruction: how fast can we introduce new numerals?  How do you introduce new numerals?

50. Introducing Numeral Identification  Write review numerals (how many times?) and new numeral (how many times?) on board.  Introduce new numeral.  (This is __. What is it?)  Discrimination practice.  (What order?)  Individual turns.

51. Introducing Numeral Identification  Why do you need to signal?  How do you signal when students are looking at the numerals on the board?  How long should you spend on this task?

52. Introducing Numeral Writing  When can you introduce numeral writing?  Rate of introduction?  What are the stages of introduction (scaffolding)?  What is numeral dictation? What order do you dictate numerals?  How do you correct student errors?

53. Introducing Symbol Identification and Writing + - =  How do you introduce symbols?

54. Introducing Equation Reading and Writing  What is equation reading a preskill for?  When is equation reading introduced?  How do you teach equation reading?

55. Introducing Equation Writing  When is equation writing introduced?  How do you teach equation writing?  How do you correct if students write numerals out of order?

56. Numeral/Object Correspondence 1. Students identify the symbol (numeral) and write that number of lines. 2. Students count the objects and write the numeral.  Preskills for addition and subtraction using equality strategy.

57. Numeral/Object Correspondence  When can you introduce these numeration skills?

58. Numeral/Object Correspondence  Before teaching students to identify the symbol and write the lines, what preskill must students have?  See format 5.3.  What errors might students make in 5.3?

59. Numeral/Object Correspondence  Why is writing numerals to represent a set of objects important? 4 + 2 = llll ll

60. Numeral/Object Correspondence  Format 5.4 teaches students to count the objects and write the numeral.  What are the preskills?  What errors might students make?

61. Numeral/Object Correspondence  What should you do if students make a counting error?  What should you do if the students make an error in numeral identification or writing?  When do you introduce manipulatives?

62. Place Value  Reading and Writing Numerals  Column Alignment  Expanded Notation

63. Reading and Writing Teen Numerals  When is reading teens numerals introduced?  What is the order of introduction?  See format 5.5.  When are “irregular” teens introduced?  What is the rate of introduction for irregular teens?

64. Reading and Writing Teens Numerals  When is writing teens numerals introduced?  See format 5.6.  When might manipulatives be used?

65. Reading Numerals 20-99  What are the preskills?  Format 5.7

66. Writing Numerals 20-99  When is this introduced (that is—what are the preskills)?  See format 5.8.  When dictating numerals in step E, what is the example selection guideline?

67. Writing Numerals 20-99  What pattern of errors might students make (diagnosis)?

68. Remediation for Written Reversals (such as 71 for 17)  Reteach writing teens format. At the same time, reteach writing tens numbers format (without 1s in the ones place—like 31).  Then teaching writing format with minimal discriminations—21 & 12, 41 & 14, etc.

69. Reading and Writing Numerals 100-999  Reading hundreds—What are the preskills?  See format 5.9.  Sequencing: What is avoided initially? Then, what examples are used?

70. Reading and Writing Numerals 100-999  Sequencing: What is avoided initially? (0 in the tens place)  What examples are used when 0 in the tens place is included?

71. Reading and Writing Numerals 100-999 Writing hundreds numerals—Format 5.10

72. Reading and Writing Numerals 1,000-999,999  What is the sequence for introducing these numerals?  What are the example selection guidelines when zeros are introduced?

73. Column Alignment  Why is this an important skill?  See format 5.13

74. Expanded Notation  What is expanded notation?  See Format 5.14.

75. Application Exercises  Counting (p. 41) 1, 5  Symbol ID & Place Value (p. 60) 6, 9

76. CURRICULUM EVALUATION