Transcript Key Strategies for Mathematics Interventions

Key Strategies for
Mathematics Interventions
Explicit Instruction
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.
The National Mathematics Advisory
Panel defines explicit instruction as:
• “Teachers provide clear models for solving a
problem type using an array of examples.”
• “Students receive extensive practice in use of
newly learned strategies and skills.”
• “Students are provided with opportunities to
think aloud (i.e., talk through the decisions
they make and the steps they take).”
• “Students are provided with extensive
feedback.”
Explicit Instruction
The NMAP notes that this does not mean that
all mathematics instruction should be explicit.
But it does recommend that struggling students
receive some explicit instruction regularly and
that some of the explicit instruction ensure that
students possess the foundational skills and
conceptual knowledge necessary for
understanding their grade-level mathematics.
Example 1
The boys swim team and the girls swim team held a car
wash. They made $210 altogether. There were twice as
many girls as boys, so they decided to give the girls’
team twice as much money as the boys’ team. How
much did each team get?
Let b = the amount of money for the boys. Since the
girls get twice as much, they get 2b. b + 2b = 210
Step 1: Label an unknown amount
Step 2: Write an equation, then solve it.
Step 1: Draw a picture to represent what you
know.
Step 2: Write an equation, then solve it.
b equals 1/3 of $210, b = 1/3 ∙ 210
Example 2
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?
• 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?
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.
Student Thinking
• Remember that an important part of explicit
instruction is that students also need to
verbalize their thinking.
• “Provide students with opportunities to solve
problems in a group and communicate
problem-solving strategies.”
Example 3
1/2 + 1/4
Create a real-world problem that corresponds to
this.
Use fraction circles to represent this problem and
find a solution. Explain your solution to your
partner.
What did you learn about equivalent fractions?
1/2 + 1/8 3/8 + 1/4 3/4 + 3/8 3/4 + 5/8
(Let the partner explain their thinking on these.)
Example 4
3/4 ÷ 1/8
Work this problem any way you can, then
compare your answers at your table. Discuss
your method for solving.
What problems do students typically have with
your method?
Create a real-world situation that is represented
by this.
Division by a fraction has a specific meaning,
which most people need to be taught explicitly.
It is the “measurement” definition of division:
How many 1/8’s are there in 3/4? (or how many
times does 1/8 go into 3/4?)
(Recommendation 4, about the underlying
structure of word problems, ties in here.)
A visual representation is the best way to teach
about the concept of division by a fraction:
Try making visual representations for
⅘÷⅕
⅘÷⅖
⅔÷⅙
1½ ÷ ¼
¾÷⅜
⅛÷¼
Conclusions about explicit teaching
It is appropriate when…
• Some important way of looking at a problem
is not evident in the situation (what division
by a fraction means)
• A useful representation needs to be presented
(the bar model)
• A heuristic is helpful in many situations (Label
an unknown, write an equation, solve it)
What is NOT explicit teaching?
http://www.khanacademy.org/video/multiplication-6--multiple-digitnumbers?playlist=Arithmetic
Which characteristics does it address, which
does it not address?
Conclusions about explicit teaching
It may be more appropriate to let students
figure things out when…
• Remembering requires deep thought (how to
find equivalent fractions)
• The goal is about making connections rather
than becoming proficient with skills
Example 5
Lucy has 8 fish. She buys 5 more fish. How many
fish does she have then?
What are the students doing in these video
clips?
How did they learn to do this?
Practice with each other using the ten-frame
cards.
Explicit and Systematic
• Work through the section on multiplication of
the Origo Math student workbook. In what
ways does this represent a systematic
approach?
Recommendation 2
Instructional materials for students receiving
interventions should focus intensely on indepth treatment of whole numbers in
kindergarten through grade 5 and on
rational numbers in grades 4 through 8.
These materials should be selected by
committee.
What would this include?
5th Gr. Common Core Standards
• Summarize the whole number and rational
number goals for 5th grade.
Recommendation 4
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.
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)
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?
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
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
• 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
• You’re driving on a vacation. You drive at 50
mph for 7 hours. How far have you driven?
• You drive 300 miles in 6 hours. How fast were
you driving, on average? (how many miles do
you go in each hour)
• How long does it take you to drive 400 miles at
50 mph?
• You’re driving on a vacation. You drive at 50
mph for 7 hours. How far have you driven?
Counting out 7 groups of 50: d = 50 · 7 This
procedure generates the formula d = r · t
• You drive 300 miles in 6 hours. How fast were
you driving, on average? (how many miles do
you go in each hour)
Dividing 300 into 6 groups (partitive division):
r = 300/6
• How long does it take you to drive 400 miles at
50 mph?
Counting how many 50’s in 400 (measurement
division). 400/50 = t
Distance =
300 miles =
400 miles =
Rate ∙
Time
50 mph
x
7 hrs
6 hrs
50 mph
x
Transfer to problems
of the same type
Area =
40
40
Length ∙
Width
5
5
8
8
Visual Representations
• The point of visual representations is to help
students see the underlying concepts.
- Draw a picture for each of the problems.
• 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
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
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 to doing
math using pictures, and then 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:
32
21
?
• Elisa has 37 dollars. How many more dollars
does she have to earn to have 53 dollars?
37 + ___ = 53
53 ducks are swimming on a pond. 27 ducks fly
away. How many ducks are left on the pond?
First, try this with mental math.
Next, model it with unifix cubes.
Then use symbols to record what we did.
4 13
53
-27
26
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
Symbols:
• 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.
Components of Mathematical
Proficiency
• Conceptual Understanding - Comprehension
of mathematical concepts, operations, and
relations.
• Procedural Fluency - Skill in carrying out
procedures flexibly, accurately, efficiently, and
appropriately.
• Strategic Competence - Ability to formulate,
represent, and solve mathematical problems.
• Adaptive Reasoning - Capacity for logical
thought, reflection, explanation, and
justification.
• Productive Disposition - Habitual inclination
to see mathematics as sensible, useful, and
worthwhile, coupled with a belief in diligence
and one’s own efficacy.