Number Sense A Critical Foundation How do we get there How would we know

1. Number Sense A Critical FoundationHow do we get there? How would we know? Francis (Skip) Fennell Professor of Education McDaniel College Westminster, MD & Past President National Council of Teachers of Mathematics Pittsburgh, PA Public Schools August 31, 2009

2. More and better mathematics for all students This is an imperative for America�s competitiveness.

3. THE COMPETITION THING

4. First things first! �A strong K�12 mathematics education for all students is important for our nation�s economic stability, future national security, and workforce productivity.�

6. MATHEMATICS FOR ALL STUDENTS NCTM, this school district, and everyone I talk to supports programs and practices that encourage the acquisition of high-level mathematics skills by a wide range of overlapping populations: children of poverty, English language learners, urban and rural students, students of all races and ethnicities, students with learning difficulties, students who are female, and students who are mathematically gifted.

8. Equity We shortchange the challenge and limit the discussion when we think equity is all about achievement gaps. Equity is about: How and when we assess Intervention Curricular Opportunities Disposition

9. All means�in the classroom Recognizing the importance of formative assessment; And the potential of intervention.

10. Continuum of Assessment Distance Immediate � informal observation or artifacts from a lesson. Close � embedded assessments and semi-formal quizzes following several activities. Proximal � formal classroom exams following a particular curriculum. Distal � criterion-referenced achievement tests such as those required by NCLB. Remote � Broad outcomes measured over time � norm referenced tests. Ruiz-Primo, Shavelson, Hamilton, and Klein (2002)

11. Reminder Assessment should not merely be done to students; rather, it should also be done for students, to guide and enhance their learning.

12. Teachers assess all the time, every day! Where does the decimal point go? 306.15 ? 75.4 = 40603448

13. All assessments are snapshots within a much larger album or record of learning.

14. Informal Assessments Observations Anecdotal Reports We know it is more informative to observe a student during a mathematical activity than to grade his papers. (Freudenthal, 1973)

15. They�re all yours� Start before they have the instructions. Seemingly get distracted by movement � of any type! Oh, yeah, I get it! I�m done!

16. Write a word problem that can be solved by either addition or subtraction

18. Summing Up: Formative Assessment Aside from teacher-made classroom tests, the integration of assessment and learning as an interacting system has been too little explored. NMP and formative assessment. Glaser and Silver, 1994

19. The Challenge of Intervention To provide guidance in the creation or selection of intervention programs that support - high-quality, equitable instructional practices for all students.

21. Think about� The mathematics: Why are some mathematics topics more challenging than others, with regard to intervention? Why are some topics �quick fixes� and others long term? Differences among computational fluency, conceptual understanding and problem solving.

22. Many students benefit Struggling students (typically) Absentees Mathematically talented students, as a challenging supplement to a standard instructional program

24. All students should leave elementary school computationally fluent AND with a strong sense of number What does that mean? How do we do that?

25. 25 Principles and Standards for School Mathematics Content Standards Number and Operations Algebra Geometry Measurement Data Analysis and Probability

26. 26 Number Sense Number Meaning Relationship Magnitude Operation Sense Real Life Number Sense - Applications Howden, 1989

27. 27 In its most fundamental form, number sense entails an ability to immediately identify the numerical value associated with small quantities; �this more highly developed form of number sense should extend to numbers written in fraction, decimal, and exponential forms. �poor number sense interferes with learning algorithms and number facts and prevents use of strategies to verify if solutions to problems are reasonable.

28. 28 Do you have a sense of number? Is 4 x 12 closer to 40 or 50? How many paper clips can you hold in your hand? If the restaurant bill was $119.23, how much of a tip should you leave? How long will it take to make the 50 mile drive to Washington, D.C.? If a 10-year old is 5� tall, how tall will the child be at age 20?

29. Computational Fluency Having and Using Efficient and Accurate Methods for Computing 29

30. Policy and Political Issues Number sense is developed! Number sense includes automaticity! Where does this fit in any state�s curricular standards? This is more than whole numbers!

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32. Counting Rote Counting Rational Counting Counting Strategies Counting On Counting Back Skip Counting

36. Five Frames

37. Ten Frames

44. 100 Chart Puzzles

45. 45

47. Subtraction Removal or take away Whole - part = part Comparison comparing parts Completion part + ? = whole; whole - part = missing part

49. Fact Acquisition Children should attempt to memorize facts only after understanding is attained. Children should participate in drill activities with the intent to develop fluency. Remembering should be emphasized. Drill lessons should be short and occur almost every day. Previously learned facts should be reviewed. Records of progress should be kept. Encourage and praise students for remembering!

50. 50 100 100 is a big number when it�s: 100 is a small number when it�s:

51. Math Wall Activities 24 73 49 Today�s Date

52. 52

53. 53 On each numbered 100 chart, shade in the numbers that are... 2-digit numbers numbers containing the digit 3 numbers < 60 odd numbers numbers between 31 and 51 numbers where the sum of the digits = 8 WHAT NUMBER DID YOU COLOR ON EVERY CHART? Marcy Cook, 1992

54. 54 On each numbered 100 chart, shade in the numbers that are... > 30 numbers with an odd digit in the tens place even numbers numbers with 2 as a digit numbers between 45 and 95 numbers with the sum of the digits = 11 WHAT NUMBER DID YOU COLOR ON EVERY CHART? Marcy Cook, 1992

56. 56 And how about this chart?

57. WIPEOUT

58. 58

59. 59

60. 60 My number of the day* Multiply your number by 4: _____ Subtract 1: _____ What is the new number? _____ How is the new number different from your number of the day? 4x � 1 = n

61. 61 Name something that helps you attach meaning to each number below: 25 50 500 75 60 36 30

62. 62 Favorites Write 3 numbers that have some significance to your life. Exchange lists. Provide random clues for the numbers. Guess which numbers fit the clues.

63. 63 Today�s Target is 36 Try to make today�s target by: Adding 2 numbers Finding the difference of 2 numbers Multiplying 2 numbers Adding 3 numbers Multiplying 3 numbers Multiplying and subtracting YOUR own method! McIntosh, Reys, Reys, and Hope (1997)

64. 64 Number Sense Language bunch pile flock herd stack handful basket cord crowd

65. 65

66. 66 Commutative Property Multiplying by 0 Multiplying by 1 Squares Doubles - 2�s facts Nickels Facts - 5�s facts Basic Facts

67. 67 9 x 0 = 0 4 x 0 = 0 9 x 1 = 9 4 x 1 = 4 9 x 2 = 18 4 x 2 = 8 9 x 3 = 27 4 x 3 = 12 9 x 4 = 36 4 x 4 = 16 9 x 5 = 45 4 x 5 = 20 9 x 6 = 54 4 x 6 = 24 9 x 7 = 63 4 x 7 = 28 9 x 8 = 72 4 x 8 = 32 9 x 9 = 81 4 x 9 = 36

68. 68 Finding and using patterns and other thinking strategies greatly simplifies the task of learning multiplication tables. Thornton, 1978 Children need to identify individual products rapidly. Little is known about how children acquire this fluency or what experiences might be of most help.

69. 69 Boxes to multiply� Draw a rectangle to show 46 x 7 = 322

70. 70 Multiplication Properties Distributive Property of Multiplication over Addition. 8 x 14 = 8(10 + 4) = (8 x 10) + (8 x 4) 14 x 8 80 32 112

73. 73 How about 45 x 23

74. Division - Models Division as successive subtraction Nelson had 125 pieces of candy. He gave 7 pieces to each of his friends. How many friends received candy? (how many groups) Division as sharing Nelson had 125 pieces of candy. He shared his candy [equally] among his 7 friends. How many pieces of candy did each friend receive? (how many in each group)

75. 75 True or False - 818 Number of students in your school? Number of people in your town? Number of players on the team? Number of pennies in a collection? Closer to 500 or 1,000? > 500 > 750

76. 76 Nope, partial credit won�t do for many important problems, which is why we: Provide a problem solving guide which we very carefully break down into component parts and help students attack problems. Ask if students are stuck and provide a lifeline to help them read, understand, plan and solve a problem. Provide lessons involve problem solving skills and strategies. Provide many, many applications to problem solving - within typical lessons as well as problem solving application lessons. Recognize, promote, and value the obvious connection between reading, writing, language, and mathematical problem solving. Nope, partial credit won�t do for many important problems, which is why we: Provide a problem solving guide which we very carefully break down into component parts and help students attack problems. Ask if students are stuck and provide a lifeline to help them read, understand, plan and solve a problem. Provide lessons involve problem solving skills and strategies. Provide many, many applications to problem solving - within typical lessons as well as problem solving application lessons. Recognize, promote, and value the obvious connection between reading, writing, language, and mathematical problem solving.

77. 77

78. 78 Estimation � Some Thoughts Estimating Magnitude � should begin early and occur often. Children are initially uncomfortable with computational estimation. The language of computational estimation is adult language. Children seem OK with such language as they grow � experientially.

79. 79 Estimation How many 1-digit numbers are there? 2-digit numbers? 3-digit numbers? The toll road is 243 miles long. If you traveled at a speed of 61 mph, about how many hours will you be on the toll road? The height of full grown human is about 21 times the length of the middle finger. ARE YOU KIDDIN� ME!!!!!!!

80. 80 NMAP - Student Preparation The first question concerned the adequacy of student preparation coming into the Algebra I classes. The topics that were rated as especially problematic were: Rational numbers; Solving word problems, and; Basic study skills.

81. 81 Fractions are a major area of study in upper elementary school mathematics. It is time to shift the emphasis and redefine the goal of fraction instruction from learning computation rules to developing fraction operation sense (Huinker, 2002). Do we do this?

82. 82

83. 83 Oh my�

84. 84 Fraction beginnings� which one is larger, 1/2 or 1/3?

85. 85 Fraction Sorting Sort the fractions below as near: 0, �, or 1 4/7 1/7 8/9 3/5 2/3 1/10 4/8 6/11 4/5 2/12 9/12 5/12 1/8 3/8 4/9 7/14 What�s alike about all fractions near 1? Near 0?

86. 86 Partitioning How can we share eleven hoagies (aka subs) among four people? How can we share eleven hoagies (aka subs) among five people?

87. 87 Now what? There are 25 students in our class. Each student will get � of a pizza. Your job is to find out how many pizzas we should order. Be sure to show your work. How many pizzas should we order?

88. 88

89. 89 � What happens to the value of the fraction if the numerator is increased by 1? What happens to the value of the fraction if the denominator is decreased by 1? What happens to the value of the fraction if the denominator is increased?

90. In which of the following are the three fractions arranged in order from greatest to least? 2/7, 1/2, 5/9 1/2, 2/7, 5/9 C. 1/2, 5/9, 2/7 D. 5/9, 1/2, 2/7 E. 5/9, 2/7, 1/2

91. Lakers vs Nuggets Which player from the Lakers had the best shooting percentage Which player from the Lakers had the worst shooting percentage Same items for Nuggets Which players scored the most points, etc.

92. 92 Ordering Fractions Write these fractions in order from least to greatest. Tell how you decided. 5/3 5/6 5/5 5/4 5/8 7/8 2/8 10/8 3/8 1/8

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94. 94 Tell me about where 2/3 + 1/6 would be on this number line (Cramer, Henry, 2002).

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97. 97 Use Percent � Don�t Wait! Put 2/3; 0.5 and 3/4 in order from smallest to largest. It�s easy, 0.5 is 50% and 2/3 is 66%, and so it goes first 0.5, then 2/3 and then � because that�s 75%.* *response by Andy in New Approaches to Teaching the Rational Number System by Joan Moss in How Students Learn: Mathematics in the Classroom, NRC, 2005.

98. 98 Percent Benchmarks 0% 100% 50% < 10% ~25% ~75% ~90% > 50% < 50% Lefthanders in the room or class Once lived in New Jersey Been involved in education > 10 years People who were born in Pennsylvania

99. You can�t make this stuff up! The weather reporter on WCRB (a Boston radio station) said there was a 30% chance of rain. �The host of the show asked what that meant. �The weather reporter said ``It will rain on 30% of the state.'' �``What are the chances of getting wet if you are in that 30% of the state?'' �``100%.'�

100. You can�t make this stuff up Gettysburg (Pa) Outlets � July 3, 2009. 50% off sale on all purchases at the Izod store. Sign indicates 50% off the all-store sale. Patron � �well that means it�s free.� Clerk � �no sir, it�s 50% off the 50% off sale.� Patron � �well, 50% + 50% is 100% so that means it should be free.� This went on for a while. AND, there was a sign indicating 70% off for some items, meaning 70% off the 50% off original sale, which our patron would interpret as the item being free and 20% in cash!

101. 101 Concluding Thoughts Computational Fluency is critical and an area of focus � It always has been! Number sense is elusive Number sense should be nurtured � every day! A sense of number breeds confidence. Number sense is not the final chapter in a 12 chapter book! Fractions � all of �em - are numbers too!

102. A couple more things�

105. The Parent Conference from � Well, I was never good in math either? Tell me, when will my child ever use this mathematics? Effort matters � NMAP The cultural thing�

106. The World is Flat� Math and science are the keys to innovation and power in today�s world, and American parents had better understand that the people who are eating their kids� lunch in math are not resting on their laurels.

107. Outliers The 10,000 hour rule �Practice isn�t the thing you do once you�re good. It�s the thing you do that makes you good.� Rice paddies and math tests

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109. We share a responsibility This must be a concerted team effort, it�s about every child every day.