Number Sense Progressions Concepts embedded in number sense

Number Sense Progressions “Concepts embedded in number sense may be as important to early math learning as concepts of phonemic awareness”- Gerslen and Chard 1999 Using Progression Guides and Activities

Progressions Documents for the Common Core Math Standards n The Common Core State Standards in mathematics were built on progressions: n informed both by research on children's cognitive development and by the logical structure of mathematics. Working team Richard Askey (reviewer) Sybilla Beckmann (writer) Douglas Clements (writer) Phil Daro (co-chair) Skip Fennell (reviewer) Brad Findell (writer) Karen Fuson (writer) Roger Howe (writer) Cathy Kessel (editor) William Mc. Callum (chair) Bernie Madison (writer) Dick Scheaffer (writer) Denise Spangler (reviewer) Hung-Hsi Wu (writer) Jason Zimba (co-chair)

Counting and Cardinality n Several progressions originate in knowing number names and the count sequence. K. CC. 1

Pre-Counting n The key focus in pre-counting is an understanding of the concepts more, less and the same and an appreciation of how these are related. n Children at this stage develop these concepts by comparison and no counting is involved. n These concepts lay the foundation for children to later develop an understanding of the many ways that numbers are related to each other; for example five is two more than three, and one less than six.

From saying the counting words to counting out objects n Number sense begins with early counting and telling how many in one group of objects. n Students usually know or can learn to say the counting words up to a given number before they can use these numbers to count objects or to tell the number of objects. 1, 2, 3 Count to 100 by ones K. CC. 1 To count a group of objects, they pair each word said with one object. K. CC. 4 a

n Counting objects arranged in a line is easiest; - rectangular arrays (they need to ensure they reach every row or column and do not repeat rows or columns); -circles (they need to stop just before the object they started with); and -scattered configurations (they need to make a single path through all of the objects). K. CC. 5

n Remember: Only the counting sequence is a rote procedure. The meaning attached to counting is key conceptual idea on which all other number concepts are developed. Student have to be fluent enough to have enough attention to remember the number of objects that is being counted.

Activities n 1. Finding the Same Amount Give students a collection of cards with sets of number on them. Have the students pick up any card in the set and find another card with the same amount on it. Continue to find other pairs. Questions: How do you know that the cards have the same amount?

Activities n 2. Make Sets of More/Less/Same Provide students with cards with sets of 4 -12 objects, a set of small counters, and some word cards labeled More, Less, and Same. Next to each card have students make three collections of counters: a set that is more, one that is less, and one that is the same. The appropriate labels then can be placed on the sets. Have them show (Justify) how they know there are more in one group than another. Questions: How do you know five is more than four?

Activities Go to Today’s Meet for links to the resources today… http: //todaysmeet. com/MEand. CCSS 1 1. Illuminations: Okta's Rescue : http: //illuminations. nctm. org/Activity. Detail. aspx? ID=219

Number Relationships n Once children acquire a concept of counting n 1. 2. 3. 4. meaningfully…. more relationships must be created for children to develop number sense, a flexible concept of number not completely tied to counting. Spatial Relationships More and Less Than Anchors of 5 and 10 Part-Whole Relationships

From subitizing to single-digit arithmetic fluency n Students come to quickly recognize the cardinalities of small groups without having to count the objects; this is called perceptual subitizing. n Perceptual subitizing develops into conceptual subitizing—recognizing that a collection of objects is composed of two sub collections and quickly combining their cardinalities to find the cardinality of the collection

We read 7 in stages; Stage 1 Working on correspondence and counting skills- one by one

Stage 2 n Students will need to be presented small numbers that they can subitize and begin to see quickly. n Once students subitize up to 4 -5 they develop the ability to combine numbers into larger numbers

Stage 3 n 3+4=7 7=3+4 n 5+2 n 3+3+1 n Student has developed number sense through deeper understanding of quantity, number composition, different forms of a number and equality. n 3+4=34 8 -5=8

n Example: ME 88 and 94 Example: ME Green Flash Cards

Activities n Learning Patterns Provide each student with about ten counters and a whiteboard as a mat. Hold up a “dot plate” for about 3 seconds. Say “Make the pattern/draw the pattern you saw using the counters or on the whiteboard. Spend time discussing the configuration of the pattern and how many dots. Do this with a few new patterns each day. Questions: n How many dots did you see? n How did you see them? n What is a different way to see the total number of dots? http: //teachmath. openschoolnetwork. ca/documents/dotplatepatterns. VDW. pdf

Activities n Flash Cards Show a student a flashcard with, for example, 7 things in groupings of 5 and 2. Question: n How many things are there? n What helped you see how many there are?

Activities n Dice combinations Organize students into pairs. Give each pair two dice. Have students take turns to roll the dice and then say how many dots just by looking. Ask: How many dots are on the first die? How many dots on the second die? How many dots all together? Have students use calculators to keep progressive scores. The first student to a given number could be the winner. Later extend the activity to include 3 dice.

Activities Concentration: http: //illuminations. nctm. org/Activity. Detail. aspx? ID=73

From counting to counting on Being able to count forward, beginning from a given number within the known sequence (K. CC. 2), is a prerequisite for counting on. n Understanding that each successive number name refers to a quantity that is one larger K. CC. 4 is the conceptual start for Grade 1 counting on. More advanced counting-on methods are in which a counting word represents a group of objects that are added or subtracted and addends become embedded within the total 1. OA. 6 Example: ME 141, 145, 152

Activities n One-Less/More-Than Dominoes Use the dot pattern dot dominoes to play “oneless-than” dominoes. Play in the usual way, but instead of matching ends, a new domino can be added if it has the end that is one less than the end on the board. As an extension the game can be played for two less or two more and so on.

Activities n Real Counting On (ME-Secret Code Cards) Organize students into pairs. Supply each pair with number cards, a die, a paper cup, and some counters. The first player turns over the top number card and places the indicated number of counters in the cup. The card is placed next to the cup as a reminder of how many there are. The second child rolls that die and places that many counters next to the cup. Together they decide how many counters in all. A record sheet can be provided. (Number cards in the deck can be adjusted as needed)

Activities Counting on Busy Bees: http: //www. hbschool. com/activity/busy_bees/index. html How Many Under the Shell: http: //illuminations. nctm. org/Activity. Detail. aspx? ID=198 Math Expressions Find a Friend; http: //www. eduplace. com/kids/mthexp/content/egames/faf_grade 1. html

From comparison by matching n K. CC. 6 and K. CC. 7 focuses on which of two groups has more than or less than the other. n Student first learn to match the objects in the two groups to see if there any extra. n Then they begin to count the objects in each group and use their knowledge of the count sequence to decide which number is greater/less than the other. n Students learn that even if a group looks like it has more matching or counting may reveal a different result.

Activity n Make Sets of More/Less/Same Provide students with cards with sets of 4 -12 objects, a set of small counters, and some word cards labeled More, Less, and Same. Next to each card have students make three collections of counters: a set that is more, one that is less, and one that is the same. The appropriate labels then can be placed on the sets. Have them show (Justify) how they know there are more in one group than another. Questions: How do you know five is more than four?

Comparison by numbers to comparison involving addition and subtraction Comparing numbers progresses in Grade 1 to adding and subtracting in comparing situations OA. 1 n To learn that 6 and 8 are related by the twin relationship of two more or two less requires reflection. Example: ME- More and Less Daily Routine and 540

Activity The circle table seats 12 students. The rectangle table seats 8 students. How many fewer students can sit on the circle table than the rectangle table. If my class has 20 students what are the different ways we could seat them using circle tables, rectangle tables, or both? What do you think is the best way? Why?

Operations of Algebraic Thinking n Students develop meanings for addition and subtraction as they encounter problem situations in Kindergarten. n They extend these meanings as they encounter increasingly difficult problem situations in Grade 1.

n Put Together/Take Apart situations with both addends unknown play an important role in Kindergarten because they allow students to explore various compositions that make each number. K. OA. 3

Part-Whole Relationships n The ability to think of number in terms of parts is a major milestone in the development of number. n Spatial Relationships n More and less than concepts n Part-Whole –most important according to Van de Walle (2006) n Anchors or Benchmarks n Nothing in counting will cause a child to focus on the fact that it could be made of two parts.

Activity n Parts Provide children with one type of material, such as connecting cubes, squares, beans or colored paper. The task is to see how many different combinations for a particular number they can make using two parts. You may also see if the students are able to do more than two parts. Each different combination can be displayed.

Part-Whole Relationships n "Missing Part" activities provide maximum reflection of the combinations for a number ME: The Number Grabber page 275

Activity n Covered Parts A set of counters equal to the target amount is counted out, and the rest are put to the side. One child places the counters under a cup. The child then pulls some in to view. For example, if 6 is the whole and 4 are showing, the other child says, “Four and two is six”.

Working within 5 n Focusing attention on small groups in adding and subtracting situations can help students move from perceptual subitizing to conceptual subitizing in which they see and say the addends and the total

Working within 5 n Students will generally use fingers for keeping track of addends n *it is important that students in Kindergarten develop rapid visual and kinesthetic recognition of numbers to 5 on their fingers. ME- page 47

Activities We want to help children relate a given number to other numbers, specifically 5 and 10. The five frame really focuses on the relationship to 5 as an anchor for other numbers.

Activities n Five Frame Tell About Have children show a number on the five frame. Explain that only one counter can be in each section of the five frame. Questions n What can you tell us about # from looking at your mat? n What they observe will different a great deal form child to child. For example, with four counters, a child may say, “it’s two and two”, “it is one less than 5”, “one more counter would be 5” and other answers n Focus attention on how many more are need to make 5 or how far away from 5 a number is. n Try this activity with numbers between 5 -10 as well by filling the five frame and putting extras underneath the five frame. In discussion, focus attention on these larger numbers as 5 and some more “ 8 is 5 and 3 more

Activities Five Frame Activities: http: //illuminations. nctm. org/Activity. Detail. aspx? ID=74

Working with 10 n Composing and decomposing numbers from 11 to 19 into ten ones and some (further ones) is a vital first step children must take toward understanding baseten notation for numbers greater than 9. n The 10 frame is the most important model for developing the relationship between numbers n How students use a ten-frame provides insight into students' current number sense concept development.

Activities n Ten Frame Tell About- Ten Frames More/Less Show a number of counters on a ten frame. Ask what is two more, or one less, and so on. Now add a filled ten-frame and repeat the questions. Have students justify their answers.

Activities n Ten Frames Flash counters on a ten frame and ask students to write number sentences for the groups that they see. For example: I see 4 + 2 or I see 3 + 2 +1

Activities Ten Frame Activities: http: //illuminations. nctm. org/Activity. Detail. aspx? ID=75

First Grade Progression of OA n 1. Represent and solve new problem types n 2. Representing and solving subtypes for all unknowns in all three types n 3. Using Level 2 and Level 3 methods to extend addition and subtraction beyond 10, to problems within 20.

1. OA. 1

Activity The circle table seats 12 students. The rectangle table seats 8 students. How many fewer students can sit on the circle table than the rectangle table. If my class has 20 students what are the different ways we could seat them using circle tables, rectangle tables, or both? What do you think is the best way? Why?

The Mathematical Tasks Framework Teacher gives a strategy… TASKS as they appear in curricular/ instructional materials as set up by the teachers as implemented by students No longer have a H. C. Task Student Learning Stein, Smith, Henningsen, & Silver, 2000, p. 4

Factors Associated with the Maintenance and Decline of High-Level Cognitive Demands Decline n n n Routinizing problematic aspects of the task Shifting the emphasis from meaning, concepts, or understanding to the correctness or completeness of the answer Providing insufficient time to wrestle with the demanding aspects of the task or so much time that students drift into off-task behavior Engaging in high-level cognitive activities is prevented due to classroom management problems Selecting a task that is inappropriate for a given group of students Failing to hold students accountable for high-level products or processes Maintenance n n n n Scaffolding of student thinking and reasoning Providing a means by which students can monitor their own progress Modeling of high-level performance by teacher or capable students Pressing for justifications, explanations, and/or meaning through questioning, comments, and/or feedback Selecting tasks that build on students’ prior knowledge Drawing frequent conceptual connections Providing sufficient time to explore Stein, M. K. , Smith, M. S. , Henningsen, M. A. , & Silver, E. A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development. New York, NY: Teachers College Press.

Adding Using Level 2 and Level 3 methods to extend addition and subtraction beyond 10, to problems within 20. 1. OA. 6 Kilpatrick, Swafford, and Fidell (2001)

Activity n Addition Strategy: Make 10 on the Ten-Frame Give students a mat with two ten-frames. Use the green flash cards from Math Expressions. The students should first model each number in the two ten-frames and then decide on the easiest way to show (with out counting) what the total is. The obvious choice is to move counters into the frame showing either 8 or 9. Get students to explain what they did. Focus especially on the idea that 1 or 2 can be taken from the other number and put with the 9 or 8 to make 10 and whatever is left.

Subtraction Using Level 2 and Level 3 methods to extend addition and subtraction beyond 10, to problems within 20. ME- Page 469 Kilpatrick, Swafford, and Fidell (2001)

Activity n Back Down Through the Ten Frame Start with two ten-frames. One filled completely and the other partially filled. For 13, for example, discuss what is the easiest way to think about taking off 4 counters or 5 counters. Repeat with other numbers between 11 and 18. Have student write or say the corresponding fact.

Break: Next -Correlating Math Expressions and the CCSS

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CCSS Math Expressions Analysis Purpose of Tool 1 n Determine the extent to which the Core Content Standards for Mathematics are included in the mathematics curriculum materials-Math Expressions n Determine the extent to which Core Content Standards for Mathematics are sequenced appropriately in the mathematics curriculum materials

CCSS Math Expressions Analysis n The CCSSM specifies that “mathematical understanding and procedural skill are equally important and both are assessable using mathematical tasks of sufficient richness. ” (p. 4). n To help reviewers capture this richness in the curriculum materials, two lenses are used: coverage and balance.

Rubric

CCSS Math Expressions Materials Analysis Tool 1 n Coverage refers to the degree to which the curriculum materials attend to the content of a particular standard.

CCSS Math Expressions Materials Analysis Tool 1 n Balance addresses the degree to which the mathematics content is developed with a balance between mathematical understanding and procedural skill in ways that are consistent with the standard.

CCSS Math Expressions Materials Analysis Tool 1

Energizer: What should first graders know about the number 8?

What should first graders know about the number 8? n Skills -count to 8 -count 8 objects and know that the last number word tells how many -write the numeral 8 -recognize the numeral 8 Relationships -more and less by 1 and 2 (8 is one more than 7, one less than 9, two more than six, two less than ten, -Spatial patterns -Anchors of 5 and 10 (5 and 3, two away from ten) -Part-whole relationships (2 and 6) Other relationships - doubles 4 and 4 -Relationship to the real world (8 years old, 8 inches, 8 is one more that the 7 days of the week)

Number in Base Ten Progressions n Students’ work in the base-ten system is intertwined with their work on counting and cardinality, and with the meanings and properties of addition and subtraction. n Work in the base-ten system relies on these meanings and properties, but also contributes to deepening students’ understanding of them.

Number in Base Ten Progressions n Your task 1. Using the NBT Progressions Document- create a graphic of the progression from Kindergarten to First Grade in NBT- *note important concepts to remember 2. Go to Today’s Meet and go to the NBT Study link with a partner. Navigate through the videos and discussion questions. 3. When time is up- Small group discussions

Activity n High Cognitive Demanding Task Finding Worms

NBT Math Expressions and CCSS Analysis Purpose of Tool 1 n Determine the extent to which the Core Content Standards for Mathematics are included in the mathematics curriculum materials-Math Expressions n Determine the extent to which Core Content Standards for Mathematics are sequenced appropriately in the mathematics curriculum materials