1. Goals for today Share “what works” with each other
Possibly learn some new things about 2nd and 3rd grade math
Get new resources and do some planning
Go inside other’s classrooms
Think about how to provide intervention support as needed (“Daily 5”)

2. 3 sessions Addition and subtraction (today)
Multiplication and division (Mar. 20)
Measuring length, geometric shapes, working with time and money (Apr. 17)

3. Big Ideas about 2-3 Math
Addition and subtraction fluency builds on a strong foundation of solving structured problems and using mental math.
Multiplication fluency is rooted in 2nd grade “skip counting” and 3rd grade mental math.
Great resources are available for early numeracy activities.
Diagnostic assessments can help students who are behind their classmates.

4. Key Teaching Strategies
Teach the underlying structure of word problems to promote problem solving ability
Use visual representations often
Verbalize your thought processes when explaining procedures
Have students verbalize their thinking to each other by explaining what they’re doing and why
Provide lots of practice with feedback
Use frequent cumulative review
Emphasize reasoning, looking for patterns, and problem solving

5. Framework Resources Cognitively Guided Instruction
Add/Subtract Situations – Strategies – Fluency
Tens and Ones – Place Value – Multi-digit Operations
Van de Walle and Lovin
Kathy Richardson
Grayson Wheatley
Elementary Math Resources wiki
Georgia and New York lesson plans
Resources

6. Classroom Visits Sharing and Planning Teaching Channel Number Talks
Eventually your own classrooms??
Always share
Keep track of “What I Want to Try”
Make and take – Daily 5
Sharing and Planning

7. CCSS Collaborative Cards
Carla is packing treat bags for Halloween. Each treat bag has to have exactly the same items in it or the neighborhood kids will complain! She has 99 Hershey’s Kisses and 108 Jelly Beans.
a) What is the greatest number of treat bags she can make if she wants to use all the treats she has? Explain.
b) How many of each kind of treat is in one bag?

8. In your class, how many can…
Add and subtract single digit numbers fluently?
Add multi-digit numbers using strategies?
Solve word problems involving single-digit or multi-digit numbers?
Fluently add two- or three-digit numbers without regrouping?
Fluently add two- or three-digit numbers with regrouping?
We all recognize that children are at different levels of math learning in any given classroom. We’ll talk about mathematics teaching for the whole class as well as teaching interventions for struggling learners.

9. Purpose of these sessions
Purpose: Develop professional knowledge – deep content understanding and ability to use effective teaching strategies
Outcome: Responsive teaching, leading to greater readiness for the next grade

10. 2nd grade critical areas
(1) extend understanding of base-ten notation;
(2) build fluency with addition and subtraction;
(3) use standard units of measure; and
(4) describe and analyze shapes.

11. 3rd grade critical areas
(1) develop understanding of multiplication and division and strategies for multiplication and division within 100; (2) develop understanding of fractions, especially unit fractions (fractions with numerator 1); (3) develop understanding of the structure of rectangular arrays and of area; (4) describe and analyze two-dimensional shapes.

12. CCSS – Add and subtract
K: Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
K: Fluently add and subtract within 5.
1: Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations.
1: Add and subtract within 20, demonstrating fluency for addition and subtraction within 10.
2: Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

13. Children’s Mathematics
Cognitively Guided Instruction
Carpenter, Fennema, Franke, Levi and Empson

14. What operation is this?
Steven had 4 toy cars. He wanted 9. How many more toy cars would Steven need to have 9 altogether?
Show how a kindergarten or 1st grade student might solve this.

15. Modeling the Action
Liz had 8 cookies. She ate 3 of them. How many cookies does Eliz have left?
Liz has 3 marbles. How many more marbles does she need to buy to have 8 marbles?
Liz has 3 fish. Tom has 8 fish. How many more fish does Tom have than Liz?

16. Rachel’s Problems
Try each of the problems. Think about how students might model the action in the problem.
Discuss your solutions with a partner.
As you watch the video, think about which problems seem harder for Rachel.

17. Basic assumptions about children’s learning of mathematics
Very young children know how to solve math problems.
Children develop mathematical understanding and acquire fluency with whole number computation by solving a variety of problems in any way that they choose.
Children learn more advanced computational and problem solving strategies by watching their classmates solve problems.

18. Problem Types
solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions
CCSS
CGI
Adding to, putting together
Join
Taking from, taking apart
Separate
Comparing
Compare
Part-Part-Whole

19. Where is the unknown?
1. Lucy has 8 fish. She wants to buy 5 more fish. How many fish would Lucy have then?
3. Janelle has 7 trolls in her collection. How many more does she have to buy to have 11 trolls?
2. TJ had 13 chocolate chip cookies. At lunch she ate 5 of those cookies. How many cookies did TJ have left?
6. 11 children were playing in the sandbox. Some children went home. There were 3 children still playing in the sandbox. How many children went home?
4. Max had some money. He spent $9 on a video game. Now he has $7 left. How much money did Max have to start with?

20. Problem Types - Action
Result Change Start Unknown Unknown Unknown Join =  5 +  = 7  + 2 = 7 Separate 8 – 3 =  8 –  = 5  – 3 = 5

21. No-action problems Part-part-whole problems
6 boys and 4 girls were playing soccer. How many children were playing soccer?
10 children were playing soccer. 6 were boys and the rest were girls. How many girls were playing soccer? 10
Write number sentences for each of these. How do they differ? 6

22. No-action problems Comparison problems
Willy has 12 crayons. Lucy has 7 crayons. How many more crayons does Willy have than Lucy?
Write number sentences for each of these. How do they differ?

23. Are some more difficult?
There are 14 hats in the closet. 6 are red and the rest are green. How many green hats are in the closet?
14 birds were in a tree. 6 flew away. How many birds were left?
For 1st graders, yes.

24. Try this twice a week
Present a problem to the whole class, let them work on it individually, then have several students present their approaches.
For older children, use some two-digit problems that don’t require regrouping and some that do.
Keep track of students’ solutions.

25. Solution Strategies Direct modeling of the action in the problem
Counting strategies
Derived facts
Fluency
The structure of a problem determines how difficult it is for children to solve and determines their initial solution strategies.
Video of children using counting and derived facts.
Watch the video of several children using counting and derived facts.

26. Effects of CGI
Teachers prepared in CGI spend more time having their students solve problems, listen more to their students, and are more likely to expect students to find multiple solution strategies to problems than teachers who are not prepared in CGI (Carpenter, Fennema, Peterson, Chiang, & Loef, 1989).
CGI also results in improved performance by primary-grade students on both standardized and problem-solving tests (Carpenter et al., 1989; Fennema, Carpenter, & Peterson, 1989; Peterson et al., 1989).

27. Fluency with “math facts”
The use of manipulatives, counting and derived-fact strategies eventually grows into knowledge of most math facts.
Explicit instruction on strategies can be helpful for building math facts that haven’t come naturally through problem solving.

28. Explicit strategy instruction
For addition and subtraction, children can derive unknown facts using a combination of
anchor sums to 10 (6 + 4, 7 + 3, 8 + 2) – (7 + 6 can be derived by knowing that is 10, and 3 more is 13)
doubles (7 + 6 can be derived by knowing that is 14, so the answer is one less)
Number Talks 6+8

29. sums to 5
If you know the sums to 5, like 3+2, you can find other sums like 3+4, because 3+4 = plus 2 more
sums to 10
If you know that 6+4 = 10, then you can figure out 6+5, because it’s 1 more than 6+4
doubles plus one
6+7 = (6+6) + 1 more
doubles plus two
7+9 = (7+7) + 2 more
nines
Add ten then subtract 1

30. Fluency – Practice and Drill
“Practice” refers to lessons that are problem- based and that encourage students to develop flexible and useful strategies that are personally meaningful. “Drill” is repetitive non-problem-based activity to help children become facile with strategies they know already in order to internalize (remember) the fact combinations.
From Van de Walle, Elementary and Middle School Mathematics: Teaching Developmentally

31. Practice
Sum Search
Math Squares
Always follow the small group work with a whole class discussion where students explain their methods.
Number Talks 8+6

32. Kathy Richardson, Developing Number Concepts Bk. 2, p. 136-137

33. Drill
Once strategies are learned, students can often be induced into remembering combinations by playing competitive games that require quick recall of number combinations, such as “addition war” or “subtraction war” or games where pieces are moved around game boards by rolling dice and calling out sums or differences.

34. Vary the drill http://www.fun4thebrain.com/addition.html
Print triangle flash cards

35. Summarize and Plan
What are the critical elements of 2nd grade math instruction (and 3rd grade practice and intervention) that support children’s development of single-digit adding and subtracting?
Word problems – various types that grow in complexity
Children’s use of strategies to solve them – developmental
Number Talks to build mental math and further develop computation strategies
Games for practice that reward quick recall
Does your curriculum emphasize these things?
What are your favorite games for developing fluency?

36. CCSS – Multi-digit add/subt
1: Add within 100, including adding a two-digit number and a one digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten. 2: Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

37. CCSS – Multi-digit add/subt
2: Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds. 3: Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. (A range of algorithms may be used.)

38. A classroom view
Counting Collections to 100

39. Place value
K.NBT.1 Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (such as 18 = ); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones. 1.NBT.2 Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases: a. 10 can be thought of as a bundle of ten ones — called a “ten.” b. The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones. c. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

40. 2.NBT.1 Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases: a. 100 can be thought of as a bundle of ten tens — called a “hundred.” b. The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

41. Another classroom
Making Math Fun with Place Value Games

42. Multi-digit Progression
Mental strategies
Place value representations (e.g. base 10 blocks, pictures)
Algorithms

43. Multi-digit Problems Separating, result unknown
Peter had 28 cookies. He ate 13 of them. How many did he have left? Write this as a number sentence: 28 – 13 = ____
There were 51 geese in the farmer’s field. 28 of the geese flew away. How many geese were left in the field?
Comparing two amounts (height, weight, quantity)
There are 18 girls on a soccer team and 5 boys. How many more girls are there than boys on the soccer team?

44. Multi-digit Problems Part-whole where a part is unknown
There are 23 players on a soccer team. 18 are girls and the rest are boys. How many boys are on the soccer team?
Distance between two points on a number line (difference in age, distance between mileposts)
How far is it on the number line between 27 and 42?

45. Multi-digit Problems
There were 51 geese in the farmer’s field. 28 of the geese flew away. How many geese were left in the field?
There were 28 girls and 35 boys on the playground at recess. How many children were there on the playground at recess?
Misha has 34 dollars. How many dollars does she have to earn to have 47 dollars?
Strategies? Counting single units. Direct modeling with tens and ones. Invented algorithms: Incrementing by tens and then ones, Combining tens and ones, Compensating.
Categorize each of these.
Mental Math

46. Base Ten Concepts Using objects grouped by ten:
There are 10 popsicle sticks in each of these 5 bundles, and 3 loose popsicle sticks. How many popsicle sticks are there all together?
Students’ strategies?
The extension: The teacher puts out one more bundle of ten popsicle sticks and asks students “Now how many popsicle sticks are there all together?” What strategies would students use to answer this?

47. Development of Algorithms
The C-R-A approach is used to develop meaning for algorithms.
Without meaning, students can’t generalize the algorithm to more complex problems.

48. Practice? Drill?
What would be good kinds of practice for double digit work?
Good kinds of drill?

49. Typical Learning Problems

52. Create your own
You can differentiate instruction appropriately for your students by adjusting the numbers in the problems.
Try this twice a week before our next session.
Come back with stories to tell!
You can also create learning centers around these problems, number strategy games and fluency practice. When children work at centers during “Math Daily 5” you can bring a small group together who needs additional support from you.

53. Summarize and Plan
What are the critical elements of 2nd grade math instruction (and 3rd grade practice and intervention) that support children’s development of multi-digit adding and subtracting?
Word problems – same types, larger numbers
Children’s use of strategies to solve them – developmental
Place value understanding
C-R-A
Does your curriculum emphasize these things?
Do your students have both conceptual understanding and procedural skill?