Presented by: Margaret Adams Melrose Public Schools 1

 Define number sense and its components.  Describe how number sense develops through the grades.  Identify student’s misconceptions in number sense.  Name the steps for fact fluency. Describe the role of number sense in fact fluency. 2

 What is Number Sense?  Components of Number Sense  Fact Fluency and Number Sense 3

 Think about how you learned to compute in elementary school.  What was the instruction like?  What sorts of things did you do? 4

Definition in your own wordsFacts/characteristics ExamplesVisual Number Sense 5

 Make a list of all the important ideas that you think children should know about the number 8 by the time they finish first grade. 6

Riddle me this and riddle me that You can solve riddles…just like that! I am a prime number. I am an odd number. I am more than 6. My digits add up to 4. What number am I??? 7

Mystery Number If you count by fives, I have two left over. I am a multiple of 7. The sum of my digits is an even number. What’s the Mystery Number?? I am a composite number. 8

9

 Basic Definition: Understanding of what numbers are and how they relate to each other.  Example: ◦ 6 is half of 12 ◦ It’s also 3 doubled ◦ 1/3 of 18 ◦ 2 sets of 3 ◦ 3 sets of 2 ◦ 1 more than 5 ◦ 1 less than 7 10

 “Number sense is an emerging construct that refers to a child’s fluidity and flexibility with numbers, the sense of what numbers mean and an ability to perform mental mathematics and to look at the world and make comparisons.” ◦ Russell Gersten, David Chard 11

Number sense as a “good intuition about numbers and their relationships. It develops gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms.” Howden (1989) 12

1. Develops meaning for numbers and operations ◦ Connects numerals with situations from life experiences. ◦ Knows that numbers have multiple interpretations. ◦ Understands that number size is relative. ◦ Connects addition, subtractions, multiplication, and division with actions arising in real-word situations. ◦ Understands the effects of operating on numbers. ◦ Creates appropriate representations for operations. 13

2. Looks for relationships among numbers and operations. ◦ Decompose or breaks apart numbers in different ways. ◦ Knows how numbers are related to other numbers. ◦ Understands how the operations are connected to each other.  Multiplication/Division-Inverse  Addition/Subtraction-Inverse  Multiplication-Repeated Addition  Division-Repeated Subtraction 14

2. Looks for relationships among numbers and operations. ◦ Decompose or breaks apart numbers in different ways. ◦ Knows how numbers are related to other numbers. ◦ Understands how the operations are connected to each other.  Multiplication/Division-Inverse  Addition/Subtraction-Inverse  Multiplication-Repeated Addition  Division-Repeated Subtraction 15

3. Understands computation strategies and uses them appropriately and efficiently. ◦ Correctly performs the steps in an algorithm and can discuss why the algorithm works. ◦ Makes a conscious effort to complete calculations using prior knowledge and simpler calculations. ◦ Often uses a variety of calculation strategies, even when completing calculation involving the same operation. ◦ Chooses appropriate calculation technique to obtain exact answers and estimate. ◦ Calculates with accuracy and relative efficiency 16

4. Makes sense of numerical and quantitative situations. ◦ Expects numerical calculations to make sense. ◦ Seeks to understand relationships among quantities in real-world situation. ◦ Assesses whether the result of a calculation makes sense in the context of the numbers and real-world quantities involved. 17

 Research indicates that early number sense predicts school success more than other measures of cognition like verbal, spatial, or memory skills or reading ability. 18

Numeration Quantity/Magnitude Base Ten Equality Form of a Number ProportionalReasoning Algebraic and Geometric Thinking Components of Number Sense © 2007 Cain/Doggett/Faulkner/Hale/NCDPI Language “good intuition about numbers” 19

Quantity and Magnitude “good intuition about numbers” First three components of number sense are anchors for building a rich understanding of math. 20

 Quantity - The physical amount of something.  Magnitude - Quantity in relation to other quantities 21

 Math is not about numbers it is about quantity.  The physical reality of the mathematics that we model with symbols and the number line (how much, how far, how big, how bright, etc.).  Virtually all mathematical topics can be modeled for students using quantity as a core communicator. 22

 For instance-often times we will teach number as memorization rather than students deeply understanding sets.  Can our students count to 100 rather than understanding the quantity and magnitude first? 23

 “C-A-T” isn’t really a cat.  This is not a cat either. It is, clearly, just a picture of a cat. 24

 This is the number 8. The numeral is not eight just a C-A-T is not a cat.  Is this eight? It is not eight. It is a picture of 8. It represents the quantity nature of 8. Teach “number comprehension.” Teach the “eightness” of eight. 25

Help children “see” numbers in a way that will help them understand the compositions and decompositions of number. 26

This is a critical skill and may lay underneath early math number sense difficulties with addition and subtraction. “instantly seeing how many” Most humans can only subitize collections of 4 or 5. After 5, we must combine smaller numbers to comprehend the collection. Most humans can only subitize collections of 4 or 5. After 5, we must combine smaller numbers to comprehend the collection. 27

28

8 - 5 = = 7 29

= - 30

Different Forms of a Number-- Linking to Magnitude to Number Lines Number Worlds Griffin 31

 In reading, we develop context for students through experiences, practice, and text analysis. In early math, the number line is the context.  It is particularly important for the student to associate magnitude to the NUMBER LINE.  The idea that moving in a certain direction implies an increase in quantity. (Griffin) 32

 How large is eight?  Where does it live on the number line?  How much larger is 8 than 7?  How does 8 relate to 5 or 10? 33

Makes Ten Left Over lesson: valerie faulkner 34

Making 10: Facts within 20 + Makes Ten Left Over 8(2 + 3) (8 2) ten and 3 ones (13) lesson: valerie faulkner 35

Making 10: Facts within 20 + Makes Ten Left Over 8(2 + 3) (8 2) ten and 3 ones (13) lesson: valerie faulkner 36

Making 10: Facts within 20 + Makes Ten Left Over 8(2 + 3) (8 2) ten and 3 ones (13) lesson: valerie faulkner 37

 gle-count-a-game-of-number-sense gle-count-a-game-of-number-sense 38

Numeration First three components of number sense are anchors for building a rich understanding of math. “good intuition about numbers” 39

 It is the word, symbol, and the visualization.  numeration (noun) ◦ the action or process of calculating or assigning a number to something. ◦ a method or process of numbering, counting, or computing. 40

Geary and Hoard, Learning D isabilities in Basic Mathematics from Mathematical Cognition, Royer, Ed.  1-1 Correspondence  Stable Order  Cardinality  Abstraction  Order-Irrelevance Gellman and Gallistel’s (1978) Counting Principles 41

 Knowing the counting sequence is a rote procedure.  The meaning attached to counting is the key conceptual idea on which all other number concepts are attached. 42

 Verbal Counting has two separate skills 1. Child must be able to produce the standard list of counting words in order: “One, two, three, four…” 2. A child must be able to connect this sequence in a one-to-one correspondence with objects in a the set being counted. Each object must get one and only one count. 43

 The cardinality principle is an understanding that the last count word indicates the amount of the set. 44

 The ability to count on is a “landmark” on the patch to number sense. 45

 Three relationships students must understand: 1. One and two more, one and two less 2. Anchors or “benchmarks” of 5 and 10- Because 10 plays such a large role in our numeration system and because two fives make up 10, it is very useful to develop relationships for the numbers 1 to 10 connected to the anchors of 5 and Part-part-whole relationships- to conceptualize a number as being made up of two or more parts is the most important relationship that can be developed about numbers. 46

 In Pre-K, the student should know how to: ◦ Count to 8 (know the number words and their order). ◦ Count 8 objects and know the last word tells how many. ◦ Write the numeral 8. ◦ Recognize and read the numeral 8. 47

 Using their number sense, the student should be know… ◦ More and less by 1 and 2-8 is more than 7, one less than 9, two more that 6, and two less than 10 ◦ Patterned set for 8 ◦ Anchors to 5 and 10: 8 is 3 more than 5 and 2 away from 10 ◦ Part-whole relationships: 8 is 5 and 3, 2 and 6, 7 and 1, and so on (This includes knowing the missing part of 8 when some are hidden) ◦ Doubles: double 4 is 8 ◦ Relationships to the real world: my brother is 8 years old; my reading book is 8 inches wide. 48

 In kindergarten, students will begin to understand 16 as a set of six and a set of 10.  This work with decomposing numbers from 11 through 20 in kindergarten as seen as an essential foundation for place value.  Students should continue to extend one more than, two more than, one less than, and two less than to numbers in the teens. 49

50

Raw ScoreDevelopmental Level Score C.A. Equivalents years years years years years years years years Developmental Level Conversion Chart 51

Equality First three components of number sense are anchors for building a rich understanding of math. “good intuition about numbers” 52

 Equality is a mathematical statement of equivalence of two quantities and nothing more. 53

Presenting 7 as a collection of objects strung out one by one is appropriate initially while students are working on one to one correspondence and counting skills. 54

Once students can subitize numbers up to four and three, teachers work with students to develop their ability to combine numbers into larger numbers. Three and four together combine to make seven. This is a much more powerful understanding of seven than the one-by-one correspondence. 55

3 + 4=77=3+4 Students understand that seven has many compositions. What are some other compositions of 7? Students understand that seven has many compositions. What are some other compositions of 7? 56

57

58

Base Ten “good intuition about numbers” 59

Bridging the gap - moving children from counting to breaking up numbers

61 Students are counting by ones.

62

63 Student counts “One, two, three, four bunches of ten, and one, two, three, four, five, six, seven, eight singles.” Think about how novel this is. The student has never thought of counting a group of objects as a single item.

64

Making 10: Facts within 20 + Makes Ten Left Over 8(2 + 3) (82) ten and 3 ones (13) Moving from ones work to tens work 65

 Reflect on how strange it must sound to say “seven ones.” Students don’t say they are “seven ones” years old.  “Ten” then becomes a singular noun. 66

 Unitizing is the place value understanding that ten can be represented and thought of as one group of ten or ten individual units. 67

 When working with numbers, you can take an amount from one set and add it to another set, the total amount does not change.  Referred to as “compose and decompose” numbers. 68

 The key idea is that students practice bundling and connecting this to symbols. For example, 17 is read as seventeen, and one ten and seven ones, and modeled as one ten and seven and also seventeen singletons.  Students need to group materials so they eventually understand that one ten and ten ones are the same.

 Use Digit Correspondence Task.  Next, write 342. Have student read the number. Have the student read one more. What is the number that is 10 more? What is the number that is 10 less? Watch to see whether the student is counting on or back or if they immediately know that ten more is 352.  Next, ask student to write the number that represents 5 tens, 2 ones, and 3 hundreds. 70

 The Common Core State Standards in mathematics were built on progressions: narrative documents describing the progression of a topic across a number of grade levels, informed both by research on children's cognitive development and by the logical structure of mathematics. 71

 Read your grade level.  Annotate the text considering… ◦ the mathematical language ◦ Standards for Mathematical Practices ◦ Methods or strategies ◦ Common mistakes or misconceptions 72

 nting-by-ten-lesson nting-by-ten-lesson 73

 Identify one activity that supports teaching the base ten system for your grade level.  Make sure you are able to describe the activity and how the activity develops students’ number sense. 74

75

76

Form of a Number “good intuition about numbers” 77

 Form of the number can be defined as multiple representations of quantity, ratios, and mathematical information. 78

Different Forms of a Number Number Worlds Griffin 79

80

Proportional Reasoning “good intuition about numbers” 81

 Proportional reasoning involves a multiplicative relationship between two quantities.  Proportional reasoning is one of the skills a child acquires when progressing from the stage of concrete operations to the stage of formal operations. 82

 We view proportional reasoning as a pivotal concept. On the one hand, it is the capstone of children’s elementary school arithmetic; on the other hand, it is the cornerstone of all that is to follow.” (Lesh, R., Post, T., & Behr, M. (1988) 83

 A fourth grade class needs five leaves each day to feed its two caterpillars.  How many leaves would it take to feed twelve caterpillars?  Solve this problem using a picture model and then multiplication/division. 84

Geometric and Algebraic Thinking “good intuition about numbers” It is human to seek and build relations. The mind cannot process the multitude of stimuli in our surroundings and make meaning of them without developing a network of relations. 85

 Algebraic thinking or algebraic reasoning involves ◦ forming generalizations from experiences with numbers and computation, ◦ formalizing these ideas with the use of a meaningful symbol system, ◦ and exploring the concepts of patterns and functions. 86

 How are things related physically, what information can we derive from patterns, how are the numeric patterns and the physical world related.  It is not a coincidence that Geometric and Algebraic Thinking are right next to Quantity and Magnitude.  Algebraic and Geometric concepts describe, explain and predict quantity and magnitude in the real world. 87

 14  63  45 n =n+(1-1)=(1-1)=n 88

 When skip counting, which numbers make diagonal patterns? Which makes column patterns? Can you describe a rule for explaining when a number will have a diagonal or column pattern?  If you move down two and over one on the hundreds chart, what is the relationship between the original number and the new number? These examples extend number concepts to algebraic thinking. 89

 “When will this be true?” and “Why does this work?” questions require students to generalize (and strengthen) the number concepts they are learning. 90

 The equal sign is one of the most important symbols in elementary arithmetic, in algebra, and in all mathematics using numbers and operations.  The equal sign is the principal method of representing relationships. 91

 In the following expression, what number do you think belongs in the box? 8+4= +5 How do you think students in the early grades or in middle school typically answer this question? 92

 Where does the misconception come from?  Students come to see = as signifying “and the answer is” rather than a symbol to indicate equivalence.  Subtle shift- ◦ Rather than ask students to solve a problem, ask them to find an equivalent expression and use that expression to write an equation. ◦ So, 45+61=

Numeration Quantity/Magnitude Base Ten Equality Form of a Number Proportion al Reasoning Algebraic and Geometric Thinking Components of Number Sense “good intuition about numbers” 94

95

 a_pa_kettle_math/ a_pa_kettle_math/ 96

 Is math-language language?  How can we use some of what we do in reading instruction to improve our mathematics instruction? 97

Numeration Quantity/Magnitude Base Ten Equality Form of a Number Proportion al Reasoning Algebraic and Geometric Thinking Components of Number Sense © 2007 Cain/Doggett/Faulkner/Hale/NCDPI Language 98

99

Facts with Number Sense Fact Mastery Activities Fact Fluency 100

 Develop knowledge and associations of number relationships and number sense ◦ Connections to how numbers are related both sequentially and quantitatively ◦ Number properties (commutative, identity, zero, associative)  Examples of relationship activities ◦ Plus 1 facts, fact families, ten frames, skip counting  Reduces memorization load-390 Facts! ◦ 100 basic addition, subtraction, and multiplication facts ◦ 90 division facts (division by 0 is undefined) 101

 Additive Identify Property of Zero  Commutative Property of Addition  Plus(or add) 1 Rule: “When you plus (or add) 1, you say the next number.”  Plus 2: Skip over 1 number; next even or next odd number  Plus 9: add 10 and take away; take 1 away and add 10  Doubles +1 (Associative Property) 102

103

104

105

106

 What number comes next?  When you plus or add 1, the answer is the next number or  When you plus 1 you say the next number.  Let’s try it with some problems: ◦  Let’s apply the commutative property of addition: ◦

108

109

110

 What number is 2 more than that?  Strategies ◦ Plus 1 and plus 1 more ◦ Count 2 more ◦ It’s the next even or odd number  Let’s try it with some problems: ◦  Let’s apply the commutative property of addition: ◦

112

 Think plus 10 and take away 1  Think take away 1 and plus

114

115

116

117

 Number families: 3 related numbers make 2 addition and subtraction facts ◦ 8,5,3: 5+3=8 and 3+5=8  Ten frames: Visualize/recognize sums of 10 ◦ 7+3 and and 4+6  Sums of 10 + how many more? (Associative Property) ◦ 8+3 (8+2=10 so 1 more=11) ◦ 5+7 (5+5=10 and 2 more =12) 118

 Emphasize inverse relationship to addition  Minus 0  Minus 1  Minus doubles (16-8)  Minus 2  Think addition (with sums of 10 or less)  Number families 119

 Students should move to mastery activities ONLY after number sense activities are well established.  Moving to mastery activities too quickly can cause students to revert to inefficient finger strategies. 120

 Ecourage students to continue to employ the number sense activities to retrieve answers to facts.  Encourage students to skip problems they can’t recall quickly or with new strategy. (No counting fingers.)  Goal is accuracy, fluency, and automaticity. 121

 Provide massed practice on a limited set of new facts practiced with a number sense activity  Followed by cumulative review with mastered facts  Control response time (2-3 seconds per fact)  problems is sufficient (1 to 2 minutes)  Set high criteria (95-10% accuracy) 122

 Provide cumulative review during mastery activities.  Systematically integrate the need for math facts in daily math activities.  Encourage students to use knowledge of math facts to solve problems in daily math lessons. 123

 How is math exactly like a mystery novel?  Do we really try to see the BIG picture?  How does our chapter in the novel fit the whole story? 124