Transcript Key Strategies 1-23-14

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 Knowledge
To 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 Programs
To 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
• place value
• drawings
• properties of
• strategies
operations
• relationship
between addition
and subtraction
• mentally find…
• explain the
reasoning, explain
why…
• fluently add and
subtract
• use algorithms
• Your
interpretations:
Learning Progressions
1st - 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)
Diagnostic Assessments
• See the wiki
Teaching Strategies
• Underlying structure of word
•
•
•
•
•
problems
Mathematical practices:
reasoning and problemsolving
Visual representations
Explicit teaching with practice,
feedback and cumulative
review
Use of C-R-A
Motivation
C-R-A
1. Mental strategies
2. Concrete objects
3. Visual Representations
4. Abstract symbolic
procedures (Algorithms)
Objects-Pictures-Symbols
Multi-digit Problems
1. Joining, result unknown
Our 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-whole
There 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 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 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 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?
18
?
Visual representations
23
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?
Children’s Strategies
There were 28 girls and 35
boys on the playground at
recess. How many children
were there on the playground
at recess?
C-R-A
 Strategies: See Handout
4. Abstract symbolic
Incrementing by tens and
then ones, Combining tens
and ones, Compensating.
1. Mental strategies
2. Concrete objects
3. Visual Representations
procedures (Algorithms)
Objects-Pictures-Symbols
Mathematical practices: reasoning and
problem-solving
Number Talks
A 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.
Visual representations
Explicit teaching with practice,
feedback and cumulative review
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
• Underlying structure of word
•
•
•
•
•
problems
Mathematical practices:
reasoning and problemsolving
Visual representations
Explicit teaching with practice,
feedback and cumulative
review
Use of C-R-A
Motivation
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.
Multiple approaches to proficiency
Alternative Algorithms
• Adding: Partial sums
• Subtracting: Add ten
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 decimals
The graphic shows why
0.1 x 0.1 = 0.01
1
10
1
10
1
,
100
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 decimals
Estimate:
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 = 1.4
1.4 x 0.3 = 1 x 0.3 + 0.4 x 0.3
= 0.3 + 0.12
Answer is 1.4 + 0.3 + 0.12
1.4
x 1.3
0.12
0.3
0.4
1
1.82
Multiplication with Fractions
1
6
of
2
7
or
1
2
×
6
7
1
6
of
2
7
C: use fraction circles or fraction bars
R: draw a picture
=
3
7
1
6
of
2
7
or
1
2
×
6
7
C: use fraction circles or fraction bars
R: draw a different picture (standard area model)
1
2
6
7
A: this second picture develops the algorithm × =
6
14
• What problem does this illustrate?
Middle School Examples
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?
• 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?
c
.06c
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.
Create two problems similar to the previous one
that allow students to transfer what they’ve learned
to the new problem.
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?
Underlying structure: Join problem
__ + tax = 23.95
Division of whole numbers
• Visual representation: Partitioning
• 354 photos to share among 3 children
Partitive Division
354 ÷ 3
(300 + 50 + 4) ÷ 3 = 100 + 10 + 1 r 21
100 + 10 + 1 + 7
Measurement Division
Also 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 bags
165 – 60 = 105 10 bags
105 – 60 = 55 10 bags
45 – 30 = 15
5 bags
15 – 12 = 3
2 bags
37 bags with 3 cookies left over
Division with Fractions
C-R-A?
C: Try using manipulatives to figure out:
1 1
3 ÷
2 2
R:
A: One abstract/symbolic procedure (algorithm) is to find
common denominators, then divide the numerators.
How is the problem above related to
3
1
4
1
÷
4
? Draw it.
Try these with drawings:
3
1
÷
4
2
,
8
12
÷
1
3
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:
7
3
1
4
1
2
÷ , 2 ÷
5
6
Explain how you would do this, clearly and explicitly. Assign
another problem to us and monitor our work. Provide
corrective feedback. Continue to scaffold.
3
3 1 4
2
3
÷ =
× = ×2
1
4 2
2
4
2
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 Knowledge
To 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