KEY STRATEGIES FOR MATHEMATICS INTERVENTIONS
Interventionists’ Clientele• Students who may have trouble learning at the same pace
as the rest of the class• Students who may need alternative ways of looking at the
content• Students who may have learning disabilities
Professional KnowledgeTo be effective as an interventionist, you must know:• Details of each CCSS (knowledge, skill, problem-solving)• Learning progressions for each topic• Use of diagnostic assessments• Research-based teaching strategies• Multiple approaches to proficiency
These are the learning goals for today’s session.
Intervention ProgramsTo be a good consumer, you must have the professional knowledge to judge which are adequate, and when they need to be adapted.
Many are available, few are listed on the What Works Clearinghouse.
See our wiki
Strategies work in unison• Underlying structure of word problems• Mathematical practices: reasoning and problem-
solving• Visual representations• Explicit teaching with practice, feedback and
cumulative review• Use of C-R-A• Motivation
An example Multi-digit addition and subtraction• CCSS• Learning progressions• Diagnostic assessments• Teaching strategies• Multiple approaches
CCSS• concrete models• drawings• strategies
• mentally find…• explain the
reasoning, explain why…
• place value• properties of
operations• relationship
between addition and subtraction
• fluently add and subtract
• use algorithms
• Your interpretations:
Learning Progressions1st - Joining, separating and comparing problems within 20.
Demonstrate fluency within 10.Add and subtract special cases within 100.
2nd - Fluently add and subtract within 20.Solve problems fluently within 100.
3rd - Add and subtract within 1000 using strategies and a range of algorithms.
4th - Fluently add and subtract multi-digit numbers using the standard algorithms. (up through 1,000,000)
Teaching StrategiesC-R-A1. Mental strategies2. Concrete objects
3. Visual Representations4. Abstract symbolic
procedures (Algorithms)
Objects-Pictures-Symbols
• Underlying structure of word problems
• Mathematical practices: reasoning and problem-solving
• Visual representations• Explicit teaching with practice,
feedback and cumulative review
• Use of C-R-A• Motivation
Multi-digit Problems1. Joining, result unknownOur school has 34 fish in its aquarium. The 3rd grade class bought 15 more fish to add to the aquarium. Now how many fish are in the aquarium?
2. Part-part-wholeThere were 28 girls and 35 boys on the playground at recess. How many children were there on the playground at recess?
Underlying structure of word problems
3. Separating, result unknownPeter had 28 cookies. He ate 13 of them. How many did he have left? Write this as a number sentence: 28 – 13 = ____
There were 53 geese in the farmer’s field. 38 of the geese flew away. How many geese were left in the field?
4. 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?
3. Part-whole where a part is unknownThere are 23 players on a soccer team. 18 are girls and the rest are boys. How many boys are on the soccer team?
4. Distance between two points on a number line (difference in age, distance between mileposts)
Misha has 34 (27) dollars. How many dollars does she have to earn to have 47 (42) dollars?
18 ?23
Visual representations
Children’s StrategiesThere were 28 girls and 35 boys on the playground at recess. How many children were there on the playground at recess?
Strategies: See Handout Incrementing by tens and then ones, Combining tens and ones, Compensating.
C-R-A1. Mental strategies2. Concrete objects
3. Visual Representations4. Abstract symbolic
procedures (Algorithms)
Objects-Pictures-Symbols
Mathematical practices: reasoning and problem-solving
Number TalksA classroom method for developing understanding, skillful performance and generalization
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.
Explicit teaching with practice, feedback and cumulative review
Visual representations
Recommendation 3: Instruction during the intervention should be explicit and systematic. This includes
• providing models of proficient problem solving, • verbalization of thought processes, • guided practice, • corrective feedback, and • frequent cumulative review.
Strategies are Braided
Recommendation 8. Include motivational strategies in tier 2 and tier 3 interventions.• Reinforce or praise students for
their effort and for attending to and being engaged in the lesson.
• Consider rewarding student accomplishments.
• Allow students to chart their progress and to set goals for improvement.
• Underlying structure of word problems
• Mathematical practices: reasoning and problem-solving
• Visual representations• Explicit teaching with practice,
feedback and cumulative review
• Use of C-R-A• Motivation
Alternative Algorithms• Adding: Partial sums• Subtracting: Add ten
Multiple approaches to proficiency
Practice vs. Drill• Practice usually involves word problems that draw out
strategies. Students get good at using the strategies through practice. Strategies may include algorithms.
• Drill usually doesn’t involve word problems. It is repetitive work that solidifies a student’s proficiency with a given strategy or procedure.
Typical Learning Problems
Always start by determining what the student is doing correctly.
Multiplication• Visual representations
Multiplication C-R-A• Visual representations translate to symbolic
Learning Progression
Multiplication with decimalsThe graphic shows why 0.1 x 0.1 = 0.01 or , which helps develop the algorithm. 5.NBT.7 Add, subtract, multiply, and divide decimals to hundredths, 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.
Multiplication with decimalsEstimate: 1.4 x 1.3 is somewhere between 1 and 2
Distributive Property:1.4 x 1.3 = (1.4 x 1) + (1.4 x 0.3)1.4 x 1 = 1.41.4 x 0.3 = 1 x 0.3 + 0.4 x 0.3
= 0.3 + 0.12Answer is 1.4 + 0.3 + 0.12
1.4x 1.3 0.12 0.3 0.4 1 1.82
Multiplication with Fractions of or
of
C: use fraction circles or fraction barsR: draw a picture
of or
C: use fraction circles or fraction bars
R: draw a different picture (standard area model)A: this second picture develops the algorithm
• What problem does this illustrate?
Middle School ExamplesLeroy paid a total of $23.95 for a pair of pants. That included the sales tax of 6%. What was the price of the pants before the sales tax?• Pretend you’re the students and solve this in groups of 3.
Explain your reasoning to each other. • What can you explain about your own thinking that would
help a struggling learner?• What methods can you teach explicitly that a student
might not figure out on their own?
Leroy paid a total of $23.95 for a pair of pants. That included the sales tax of 6%. What was the price of the pants before the sales tax?
Label a variable: Let c = cost of the pants.
Understand that 6% is not of the total cost, but 6% of the cost of the pants: 6% of c (.06)∙c
Write an equation: The cost of the pants c plus the sales tax (.06)∙c equals the TOTAL COST ___________
This is where your professional judgment comes in. If you tell the student what equation to write, they’ll come to depend on you to always tell them.
c .06c
Create two problems similar to the previous one that allow students to transfer what they’ve learned to the new problem.
Underlying structure: Join problem __ + tax = 23.95
Leroy paid a total of $23.95 for a pair of pants. That included the sales tax of 6%. What was the price of the pants before the sales tax?
Division of whole numbers• Visual representation: Partitioning• 354 photos to share among 3 children
Partitive Division354 ÷ 3(300 + 50 + 4) ÷ 3 = 100 + 10 + 1 r 21
100 + 10 + 1 + 7
Measurement DivisionAlso called repeated subtraction
• Our class baked 225 cookies for a bake sale. We want to put them in bags with 6 in each bag. How many bags can we make?
225 – 60 = 165 10 bags165 – 60 = 105 10 bags105 – 60 = 55 10 bags45 – 30 = 15 5 bags15 – 12 = 3 2 bags 37 bags with 3 cookies left over
Division with FractionsC-R-A?C: Try using manipulatives to figure out:
R:
A: One abstract/symbolic procedure (algorithm) is to find common denominators, then divide the numerators. How is the problem above related to ? Draw it.
Try these with drawings: ,
Explain how you would do this, clearly and explicitly. Assign another problem to us and monitor our work. Provide corrective feedback. Continue to scaffold.
Try these with the alternative algorithm: ,
Explain how you would do this, clearly and explicitly. Assign another problem to us and monitor our work. Provide corrective feedback. Continue to scaffold.
Strategies work in unison• Underlying structure of word problems• Mathematical practices: reasoning and problem-
solving• Visual representations• Explicit teaching with practice, feedback and
cumulative review• Use of C-R-A• Motivation
Professional KnowledgeTo be effective as an interventionist, you must know:• Details of each CCSS (knowledge, skill, problem-solving)• Learning progressions for each topic• Use of diagnostic assessments• Research-based teaching strategies• Multiple approaches to proficiency
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