Key Strategies for Mathematics Interventions. 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? - PowerPoint PPT Presentation
Key Strategies for Mathematics Interventions
Key Strategies for Mathematics InterventionsHeather 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.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?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.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.Being an interventionistrequires 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.
Content KnowledgeCommon Core Standards Critical areas at each grade (focus on number and operations)Common Core Standards Learning progression framework: Understanding, skillful performance, generalization
Instructional StrategiesAlong with in-depth content knowledge, both classroom teachers and interventionists need to be skillful at using proven instructional strategies. Agenda1. Review the Common Core Standards for the areas you teach2. Consider the key research-based instructional strategies as outlined in the IES Practice GuideVisual representations (C-R-A framework)Common underlying structure of word problems Explicit instruction including verbalization of thought processes and descriptive feedbackSystematic curriculum and cumulative review
Content Knowledge StandardsIn 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.Content Knowledge StandardsIn 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.11Content Knowledge StandardsIn 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.Content Knowledge StandardsIn 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.
Content Knowledge StandardsIn 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.Content Knowledge StandardsLook through the shortened versions of the standards and group them into main topics. Look for learning progressions within the topics.Learning ProgressionAdding and subtracting begins with basic understanding of number relationships:Counting on, counting backMaking five, making ten (knowing combinations to the anchor numbers of 5 and 10)
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).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, whats 520 + 215 ? (use mental math)Fluently adding and subtracting within 100 (2nd grade) and 1000 (3rd grade).
Learning ProgressionMultiplying 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
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 multiplyinguse area and array models to represent multiplication and to simplify calculations.
Learning Progressionunderstand how the distributive property works and use it to simplify calculations 15 x 8 = (10 x 8) + (5 x 8)use alternative algorithms like the partial product method (based on the distributive property) and the lattice methodbe able to identify typical errors that occur when using the standard algorithm.
Learning ProgressionTypes of KnowledgeUnderstanding conceptsSkillful performance with procedures (fluency)Generalizations that support further learning
ExamplesUnderstanding what addition and subtraction mean; understanding the concept of place valueSkillful performance of single-digit additionGeneralization of skip counting to multiplication
Key StrategiesVisual representations (C-R-A framework)Common underlying structure of word problems Explicit instruction including verbalization of thought processes and descriptive feedbackSystematic curriculum and cumulative review
What does it mean to focus intensely? And systematically.25Visual RepresentationsIntervention 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)Using the math problems, draw a picture to represent each one. 26Take 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 isnt symbols or words.
Visual RepresentationsThe 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
Using the math problems, draw a picture to represent each one. 28CRA for decomposing 5C: 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
Objects Pictures SymbolsYoung 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.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:
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:
3221?Do CRAs on paper32Elisa has 37 dollars. How many more dollars does she have to earn to have 53 dollars? (Try it with mental math.)
37 + ___ = 53
C-R-A53 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)
Show Number Talks 2nd grade multi-digit
34C-R-A53 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
Show Number Talks 2nd grade multi-digit3518 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 blocksPictures: Use an area model