Key Strategies for Mathematics Interventions

1. Key Strategies for Mathematics Interventions

2. Heather has 8 shells. She finds 5 more shells at the beach. How many shells does she have now? • Solve it. Show all your work. Explain how you solved it. • Make a drawing that helps solve it. • What kind of problem is this? • Make up another problem with the same underlying structure.

3. Heather has 8 shells. She finds some more shells at the beach. Now she has 12 shells. How many shells did she find at the beach? • Solve it. Show all your work. Explain how you solved it. • Make a drawing that helps solve it. • Is this the same underlying structure as the first problem?

4. 5 apples cost $2.75. How much do 12 apples cost? • Solve it. Show all your work. Write a reason for each step. • Make a drawing that helps solve it. • What kind of problem is this? • Make up another problem with the same underlying structure.

5. In a bag of 40 M&Ms we found 9 red ones. How many red M&Ms would be in a bag of 100? • Solve it. Show all your work. Write a reason for each step. • Make a drawing that helps solve it. • What kind of problem is this? • Make up another problem with the same underlying structure.

6. Being an interventionist requires all of the knowledge and skill of being a classroom teacher, plus more: Interventionists need to know where each child is on each learning progression. Classroom teachers need to know the content standards in detail. Interventionists also need to understand how learning builds within each topic area.

7. Content Knowledge • Common Core Standards – Critical areas at each grade (focus on number and operations) • Common Core Standards – Learning progression framework: Understanding, skillful performance, generalization

8. Instructional Strategies Along with in-depth content knowledge, both classroom teachers and interventionists need to be skillful at using proven instructional strategies.

9. Agenda 1. Review the Common Core Standards for the areas you teach 2. Consider the key research-based instructional strategies as outlined in the IES Practice Guide • Visual representations (C-R-A framework) • Common underlying structure of word problems • Explicit instruction including verbalization of thought processes and descriptive feedback • Systematic curriculum and cumulative review

10. Content Knowledge – Standards In Kindergarten, instructional time should focus on two critical areas: (1) representing and comparing whole numbers, initially with sets of objects; (2) describing shapes and space. More learning time in Kindergarten should be devoted to number than to other topics.

11. Content Knowledge – Standards In Grade 1, instructional time should focus on four critical areas: (1) developing understanding of addition, subtraction, and strategies for addition and subtraction within 20; (2) developing understanding of whole number relationships and place value, including grouping in tens and ones; (3) developing understanding of linear measurement and measuring lengths as iterating length units; (4) reasoning about attributes of, and composing and decomposing geometric shapes.

12. Content Knowledge – Standards In Grade 2, instructional time should focus on four critical areas: (1) extending understanding of base-ten notation; (2) building fluency with addition and subtraction; (3) using standard units of measure; • describing and analyzing shapes. Look through the shortened versions of the standards and group them into main topics. Look for learning progressions within the topics.

13. Content Knowledge – Standards In Grade 3, instructional time should focus on four critical areas: (1) developing understanding of multiplication and division and strategies for multiplication and division within 100; (2) developing understanding of fractions, especially unit fractions (fractions with numerator 1); (3) developing understanding of the structure of rectangular arrays and of area; (4) describing and analyzing two-dimensional shapes.

14. Content Knowledge – Standards In Grade 4, instructional time should focus on three critical areas: (1) developing understanding and fluency with multi-digit multiplication and division; (2) developing an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers; (3) understanding that geometric figures can be analyzed and classified based on their properties.

15. Content Knowledge – Standards Look through the shortened versions of the standards and group them into main topics. Look for learning progressions within the topics.

16. Learning Progression Adding and subtracting begins with basic understanding of number relationships: • Counting on, counting back • Making five, making ten (knowing combinations to the anchor numbers of 5 and 10)

17. Children solve problems with clear underlying structures by using strategies that eventually grow into fluency: • Solving simple joining and separating problems, first by counting objects, then by using strategies such as counting on, making ten, using doubles, etc., then developing automaticity. • Writing number sentences to represent problems, including ones with missing addends or subtrahends. • Fluently adding and subtracting within 5 (kindergarten), 10 (1st grade), 20 (2nd grade).

18. Adding and subtracting with two or more digits is based on an understanding of place value: • Adding tens and tens and ones and ones. Children continue to use objects, drawings and strategies (mental math) to solve multi-digit joining, separating and comparing problems as they develop proficiency with the symbolic procedures. • For example, what’s 520 + 215 ? (use mental math) • Fluently adding and subtracting within 100 (2nd grade) and 1000 (3rd grade).

19. Learning Progression Multiplying and dividing begins with repeated addition: • know that the concept of multiplication is repeated adding or skip counting – finding the total number of objects in a set of equal size groups • be able to represent situations involving groups of equal size with objects, words and symbols

20. Learning Progression • know multiplication combinations fluently (which may mean some flexible use of derived strategies) • know how to multiply by 10 and 100 • use number sense to estimate the result of multiplying • use area and array models to represent multiplication and to simplify calculations.

21. Learning Progression • understand how the distributive property works and use it to simplify calculations15 x 8 = (10 x 8) + (5 x 8) • use alternative algorithms like the partial product method (based on the distributive property) and the lattice method • be able to identify typical errors that occur when using the standard algorithm.

22. Learning Progression

23. Types of Knowledge Understanding concepts Skillful performance with procedures (fluency) Generalizations that support further learning

24. Examples Understanding what addition and subtraction mean; understanding the concept of place value Skillful performance of single-digit addition Generalization of skip counting to multiplication

25. Key Strategies • Visual representations (C-R-A framework) • Common underlying structure of word problems • Explicit instruction including verbalization of thought processes and descriptive feedback • Systematic curriculum and cumulative review

26. Visual Representations Intervention materials should include opportunities for students to work with visual representations of mathematical ideas and interventionists should be proficient in the use of visual representations of mathematical ideas. • Use visual representations such as number lines, arrays, and strip diagrams. • If visuals are not sufficient for developing accurate abstract thought and answers, use concrete manipulatives first. (C-R-A)

27. Visual Representations Take a moment to think about the visual representations of math that are used in your curriculum. Sketch as many as you can think of. These can be drawings made by children or visuals that are used for teaching. Anything that isn’t symbols or words.

28. The point of visual representations is to help students see the underlying concepts. • A typical learning progression starts with concrete objects, moves into visual representations(pictures), and then generalizes or abstracts the method of the visual representation into symbols. C – R – A

29. CRA for decomposing 5 C: How many are in this group? How many in that group? How many are there altogether? R: How many dots do you see? How many more are needed to make 5? A: 3 + ___ = 5

30. Objects – Pictures – Symbols • Young children follow this pattern in their early learning when they count with objects. • Your job as teacher is to move them from objects, to pictures, to symbols.

31. You have 12 cookies and want to put them into 4 bags to sell at a bake sale. How many cookies would go in each bag? • Objects: • Pictures: • Symbols:

32. There are 21 hamsters and 32 kittens at the pet store. How many more kittens are at the pet store than hamsters? • Objects: • Pictures: • Symbols: 32 21 ?

33. Elisa has 37 dollars. How many more dollars does she have to earn to have 53 dollars? (Try it with mental math.) 37 + ___ = 53

34. C-R-A 53 ducks are swimming on a pond. 38 ducks fly away. How many ducks are left on the pond? First, try this with mental math. Next, model it with unifix cubes. (see the C-R-A)

35. C-R-A 53 ducks are swimming on a pond. 38 ducks fly away. How many ducks are left on the pond? Then use symbols to record what we did. 4 13 53 -38 15

36. 18 candy bars are packed into one box. A school bought 23 boxes. How many candy bars did they buy altogether? • Objects: Model it with base ten blocks • Pictures: Use an area model

37. nlvm.usu.edu • Symbols:

38. You create the C-R-A • Your class is having a party. When the party is over, ¾ of one pan of brownies is left over and ½ of another pan of brownies is left over. How much is left over altogether? • Students will be at different places in the CRA learning progression.

39. Common Underlying Structure of Word Problems Interventions should include instruction on solving word problems that is based on common underlying structures. • Teach students about the structure of various problem types and how to determine appropriate solutions for each problem type. • Teach students to transfer known solution methods from familiar to unfamiliar problems of the same type.

40. Joining and Separating Problems • Lauren has 3 shells. Her brother gives her 5 more shells. Now how many shells does Lauren have? (joining 3 shells and 5 shells; 3 + 5 = ___) • Pete has 6 cookies. He eats 3 of them. How many cookies does Pete have then? (separating 3 cookies from 6 cookies; 6 - 3 = ___) • 8 birds are sitting on a tree. Some more fly up to the tree. Now there are 12 birds in the tree. How many flew up? (joining, where the change is unknown)

41. Comparing and Part-Whole • Lauren has 3 shells. Ryan has 8 shells. How many more shells does Ryan have than Lauren? • 8 boys and 9 girls are playing soccer. How many boys and girls are playing soccer? • 8 boys and some girls are playing soccer. There are 17 children altogether. How many girls are playing?

42. Multiplication • How many cookies would you have if you had 7 bags of cookies with 8 cookies in each bag? Equal number of groups • This year on your 11th birthday your mother tells you that she is exactly 3 times as old as you are. How old is she?Multiplicative comparison

43. Division • Ashley wants to share 56 cookies with 7 friends. How many cookies will each friend get? Partitive division: sharing equally to find how many are in each group • Ashley baked 56 cookies for a bake sale. She puts 8 cookies on each plate. How many plates of cookies will she have? Measurement division: with a given group size, finding how many groups

44. Addition and subtraction situations differ only by what part is unknown. Any addition problem has a corresponding subtraction problem.15 + 12 = ___ 15 + ___ = 27 • The same is true for multiplication and division.10 ∙ 8 = ___ 10 ∙ ___ = 80

46. Multiplication • A giraffe in the zoo is 3 times as tall as a kangaroo. The kangaroo is 6 feet tall. How tall is the giraffe? (write the equation) • The giraffe is 18 feet tall. The kangaroo is 6 feet tall. The giraffe is how many times taller than the kangaroo? • The giraffe is 18 feet tall. She is 3 times as tall as the kangaroo. How tall is the kangaroo?

47. 6 ∙ 3 = ___ • 6 ∙ ___ = 18 • ___ ∙ 3 = 18

48. Transfer to problems of the same type

49. Multiplication/division problems Grouping problems How many peanuts would the monkey eat if she ate 4 groups of peanuts with 3 peanuts in each group? The monkey ate 4 bags of peanuts. Each bag had the same number of peanuts in it. If the monkey ate 12 peanuts all together, how many peanuts were in each bag? (how many in each group?) The monkey ate some bags of peanuts. Each bag had 3 peanuts in it. Altogether the monkey ate 12 peanuts. How many bags of peanuts did the monkey eat? (how many groups?)

50. Rate problems A baby elephant gains 4 pounds each day. How many pounds will the baby elephant gain in 8 days? A baby elephant gains 4 pounds each day. How many days will it take the baby elephant to gain 32 pounds? A baby elephant gained 32 pounds in 8 days. If she gained the same amount of weight each day, how much did she gain in one day?