Addressing Myths and Misconceptions
with Tier II Students
Denise Schulz, Curriculum & Instruction, NCDPI
Matt Hoskins, Exceptional Children, NCDPI
MYTHS
Tier II Instruction is NOT
•
•
•
•
•
•
Slower and louder
Watered down
Rote memorization of procedure
Disconnected from conceptual understanding
Disconnected from core instruction
Shortcuts or tricks
Shepherd Problem
A shepherd is guarding his flock of 18 sheep
when suddenly 4 wolves come over the
mountain. How old is the shepherd?”
Leinwand, 2009
Teaching and
Learning
Guiding
Principles
for
School
Mathematics
Access and Equity
Curriculum
Tools and Technology
Assessment
Professionalism
Mathematics Teaching Practices
1. Establish mathematics goals to focus learning
2. Implement tasks that promote reasoning and problem
solving
3. Use and connect mathematical representations
4. Facilitate meaningful mathematical discourse
5. Pose purposeful questions
6. Build procedural fluency from conceptual understanding
7. Support productive struggle in learning mathematics
8. Elicit and use evidence of student thinking
Mathematics Teaching Practices
1. Establish mathematics goals to focus learning.
2. Implement tasks that promote reasoning and problem
solving.
3. Use and connect mathematical representations.
4. Facilitate meaningful mathematical discourse.
5. Pose purposeful questions.
6. Build procedural fluency from conceptual understanding.
7. Support productive struggle in learning mathematics.
8. Elicit and use evidence of student thinking.
Math Goals
Tasks &
Representations
What might be the math learning goals?
What representations might students use in
reasoning through and solving the problem?
Discourse &
Questions
How might we question students and structure
class discourse to advance student learning?
Fluency from
Understanding
How might we develop student understanding
to build toward aspects of procedural fluency?
Struggle &
Evidence
How might we check in on student thinking
and struggles and use it to inform instruction?
Build procedural fluency from
conceptual understanding.
Procedural Fluency should:
• Build on a foundation of conceptual
understanding.
• Over time (months, years), result in known
facts and generalized methods for solving
problems.
• Enable students to flexibly choose among
methods to solve contextual and
mathematical problems.
Baroody, 2006; Fuson & Beckmann, 2012/2013;
Fuson, Kalchman, & Bransford, 2005; Russell, 2006
Research Shows:
• Students who use memorization as their
primary strategy are the lowest-achieving
students in the world.
• Students who were achieving at high levels
were those who had worked out that numbers
can be flexibly broken apart and put together
again.
Boaler, What’s Math Got To Do With It?, 2015
Compression of Concepts vs.
Procedural Ladder
Boaler, What’s Math Got To Do With It?, 2015
Fluency builds from initial exploration and
discussion of number concepts to using
informal reasoning strategies based on
meanings and properties of the operations to
the eventual use of general methods as tools
in solving problems.
Principles to Actions (NCTM, 2014, p. 42)
Teaching Children Mathematics, August 2014
A rule that expires:
Use keywords to solve problems.
• Keywords encourage students to strip
numbers from the problem and use them to
perform a computation outside of the
problem context.
• Many keywords are common English words
that can be used in many different ways.
Karp, Bush, & Dougherty, 2014
John had 14 marbles in his left
pocket. He had 37 marbles in his
right pocket. How many marbles
did John have?
John had 14 marbles in his left
pocket. He had 37 marbles in his
right pocket. How many marbles
did John have?
Avoiding Key Words
• Key words are misleading.
• Many problems have no key words.
• The key word strategy sends a terribly wrong
message about doing mathematics.
A sense making strategy will always work.
Van de Walle & Lovin (2006)
Problem Types and Key Words
Key Word Strategies
Keywords become particularly troublesome
when students begin to explore multistep word
problems, because they must decide which
keywords work with which component of the
problem.
Karp, Bush, & Dougherty, 2014
When students are taught the underlying
structure of a word problem, they not only have
greater success in problem solving but can also
gain insight into the deeper mathematical ideas
in word problems.
Peterson, Fennema, & Carpenter, 1998)
Teaching students to distinguish superficial from
substantive information in problem also leads to
marginally or statistically significant positive
effects on measure of word problem solving.
Fuchs et al., (2003)
Start-Unknown Problem Types
• Susie has 16 marbles. This morning she gave
away 2 marbles. How many did she start
with?
• Susie has 16 marbles. Her friend just gave her
2 more marbles. How many marbles did she
start with?
How do you think students respond to these
questions if they’ve been taught a key word strategy?
Comparison Problem Types
• Susie has six marbles. This is two more than
Deante. How many marbles does Deante have?
• Deante has four marbles. This is two fewer than
Susie. How many marbles does Susie have.
• Susie has six marbles. Deante has four marbles.
How many marbles does Deante need to have the
same number of marbles as Susie?
From the Assessments:
What it is…
TIER II INSTRUCTION
Resources
LAYERING OF SUPPORT
Student Needs
Scaffolding of Tier II Instruction
Concrete
Representational
Abstract
Fisher, D., n.d.; Mercer & Mercer, 1993
National Mathematics Advisory Panel
• Explicit instruction with students who have
mathematical difficulties has shown
consistent positive effects on performance
with word problems and computation.
• This finding does not mean that all of a
students mathematics instruction should be
delivered in an explicit fashion.
(National Mathematics Advisory Panel. Foundations for Success: The Final reports of the National Mathematics
Advisory Panel, US Department of Education: Washington, DC, 2008, p. xxiii)
Explicit Instruction
(Non-Example)
Archer, 2011
Non-Responders to
Core Instruction
Instruction and Content Recommendations we will
cover today:
– Should focus intensely on in-depth treatment of whole
numbers
– Instruction should be explicit and systematic.
– Should include instruction on solving word problems that is
based on a common underlying structure.
– Instruction should include opportunities for students to
work with visual representations of mathematical ideas.
– Instruction should include motivational strategies.
Gersten, R., Beckmann, S., Clarke, B., Foegen, A., Marsh, L., Star, J. R., & Witzel, B., 2009
IN-DEPTH TREATMENT OF WHOLE
NUMBERS
WHAT CONTENT SHOULD I FOCUS TIER
II INSTRUCTION ON?
HOW SHOULD I SEQUENCE TIER II
INSTRUCTION?
Possible Content of Tier II Instruction
K-2
•
•
•
•
•
Counting
Number composition and decomposition
Meaning of addition and subtraction
Reasoning that underlies algorithms
Solving problems involving whole numbers
Word Problems:
Big Prerequisite Skills
•
•
•
•
•
Perceptual and conceptual subitizing
Rational counting
Cardinality principle
Computation within 5
Basic understanding of addition as
joining / putting together and subtraction
as taking away
Operations and Algebraic Thinking Progression Document for the Common Core State Standards
Situations for Tier II Instruction
Strategy Use: Impact of Working Memory
Geary & Hoard, 2003
Typical Development of Strategy Use
COMPONENTS OF TIER II INSTRUCTION
WHAT DOES EXPLICIT INSTRUCTION
LOOK LIKE?
HOW DO I INCORPORATE STRUCTURES?
HOW DO I USE VISUAL
REPRESENTATIONS?
Add To / Take From:
Result Unknown Situations
• Action oriented
• Represent a change in an initial state to a final
state
• Unit remains the same
• Are readily modeled through manipulative
objects, visual representations, and equations
Add To- Result Unknown
Juan had 7 plantains. He got 4 more plantains.
How many plantains does he have now?
+
Teacher Model and Think AloudConcrete
• Demonstrate a joining process
• Verbally label the unit for the start, change, and
total
• Use the term “total” instead of “sum”
• Match the demonstration to the strategy usage
the student is acquiring
• Scaffold the placement of the manipulative
objects to the rationale counting progression (if
needed)
Your Turn
Julie had 6 lollipops. Her friend gave her 7
more. How many lollipops does she have now?
Model the following strategies concretely:
• Count all
• Count on
• Make 10 strategy
Teacher Model and Think AloudRepresentational
•
•
•
•
Drawings (students acquiring level 1 strategy use)
Number lines (students acquiring level 2 strategy use)
Tens frames (students acquiring level 3 strategy use)
Part-part-whole models (students fluent with
computation)
Teacher Model and Think AloudRepresentational
Drawings
– Mirror concrete but use visual drawings
– Demonstrate a joining process
– Verbally label the unit for the start, change, and total
– Use the term “total” instead of “sum”
– Scaffold the placement of the drawings to the
rationale counting progression (if needed)
Your Turn
Julie had 6 lollipops. Her friend gave her 7
more. How many lollipops does she have now?
Model this problem using drawings and a count
all strategy
Drawing
Teacher Model and Think AloudRepresentational
Number Line
– Think aloud to correspond the start of the
problem to a point on the number line
– Think aloud to correspond to the direction of
movement on the number line
– Model movement on the number line to
correspond to the change by counting each unit
– Verbally label the start, change, and total units
– Use the term “total” instead of “sum”
Your Turn
Julie had 6 lollipops. Her friend gave her 7
more. How many lollipops does she have now?
Model this problem using the number line
0
1
2
3
4
5
6
7
8
9
10
11
12
13
0
1
2
3
4
5
6
7
8
9
10
11
12
13
Teacher Model and Think AloudRepresentational
Tens Frames
– Think aloud and demonstrate the representation of
the start of the problem in the tens frame
– Think aloud to determine a joining process
– Think aloud and demonstrate the decomposition and
re-composition of the change value to represent the
total
– Verbally label the unit for the start, change, and total
– Use the term “total” instead of “sum”
Your Turn
Julie had 6 lollipops. Her friend gave her 4
more. How many lollipops does she have now?
Model this problem using a double tens frames.
Teacher Model and Think AloudRepresentational
Part-Part-Whole Models
– Think aloud and describe the parts and whole of the model
– Think aloud and verbally label the start of the problem
– Write the numeral(s) to correspond with the value of the start and
name the units
– Think aloud and verbally label the change in the problem
– Write the numeral(s) to correspond with the value of the change in
the problem and name the units
– Think aloud and verbally label the total in the problem
– Complete the computation and write the numerals that correspond
with the total
– Verbally label the unit for the start, change, and total
– Use the term “total” instead of “sum”
Your Turn
Julie had 6 lollipops. Her friend gave her 7
more. How many lollipops does she have now?
Model this problem using a part-part-whole
model
?
6 Lollipops
7 Lollipops
Use and connect mathematical
representations.
Use and connect
mathematical
representations.
through the representations of those ideas.
Math
Because of the abstract nature of mathematics,
Teaching
Practice
people
have
access
to
mathematical
ideas
only
3
(National Research Council, 2001, p. 94)
Tasks &
Representations
What representations might students use in
reasoning through and solving the problem?
Different Representations should:
• Be introduced, discussed, and connected.
• Be used to focus students’ attention on the structure
of mathematical ideas by examining essential
features.
• Support students’ ability to justify and explain their
reasoning.
Lesh, Post, & Behr, 1987; Marshall, Superfine, & Canty, 2010;
Tripathi, 2008; Webb, Boswinkel, & Dekker, 2008
Use and connect mathematical
representations.
Illustrate, show, or work with
mathematical ideas using diagrams,
pictures, number lines, graphs, and
other math drawings.
Use concrete objects to
show, study, act upon, or
manipulate mathematical
ideas (e.g., cubes, counters,
tiles, paper strips).
Situate mathematical ideas in
everyday, real-world,
imaginary, or mathematical
situations and contexts.
Record or work with mathematical
ideas using numerals, variables,
tables, and other symbols.
Use language to interpret,
discuss, define, or describe
mathematical ideas, bridging
informal and formal
mathematical language.
Important Mathematical Connections
between and within different types of representations
Visual
Physical
Contextual
Symbolic
Verbal
Principles to Actions (p. 25) (Adapted from Lesh, Post, & Behr, 1987)
How are these solutions connected?
Proficient problem solvers
frequently use representations to
solve problems and communicate
results.
Boaler, What’s Math Got To Do With It?, 2015
Recommendations for Implementation
• Select visual representations that are appropriate for students
and the problems they are solving (Woodward et al., 2012).
• Teach students that different algebraic representations can
convey different information about an algebra problem (Star et
al., 2015).
• Make explicit connections between representations.
• Strengthen students’ representational competence by:
1.
2.
3.
Encouraging purposeful selection of representations.
Engaging in dialogue about explicit connections among
representations.
Alternating the direction of the directions made among
representations.
(NCTM, 2014).
Fluency from
Understanding
How might we develop student understanding
to build toward aspects of procedural fluency?
Procedural Fluency should:
• Build on a foundation of conceptual
understanding.
• Over time (months, years), result in known
facts and generalized methods for solving
problems.
• Enable students to flexibly choose among
methods to solve contextual and
mathematical problems.
Baroody, 2006; Fuson & Beckmann, 2012/2013;
Fuson, Kalchman, & Bransford, 2005; Russell, 2006
Take From- Result Unknown
• 8 cupcakes were on the table. I ate 2
cupcakes. How many cupcakes are on the
table now?
Teacher Model and Think Aloud:
Concrete
• Count the set and verbally label the unit
• Demonstrate taking from the set while naming
the unit
• Count the set remaining and label the unit
Your Turn
There were 6 monkeys in the tree. 3 monkeys left the
tree. Now how many monkeys are in the tree?
Model the following problem concretely using a level
one strategy.
Teacher Model and Think Aloud:
Representational
Teacher Model and Think Aloud:
Representational
There were 6 monkeys in the tree. 3 monkeys left the tree. Now how many
monkeys are in the tree?
0
1
2
3
4
5
6
7
8
9
10
11
12
13
Teacher Model and Think Aloud:
Representational
Teacher Model and Think Aloud:
Representational
6 Monkeys
3 Monkeys
?
How are these solutions connected?
Put Together/Take Apart
Total Unknown
3 red apples and 2 green apples are
on the table. How many apples are
on the table?
Put Together/Take Apart
Total Unknown
What distinguishes an Add to-Result
Unknown problem from a Put
Together-Total Unknown problem?
Put Together/Take Apart
3 bananas + 2 apples =
Put Together/Take Apart
5 banapples?
The Big Idea: The Unit
The Big Idea: The Unit
The Big Idea: The Unit
The Big Idea: The Unit
Teacher Model and Think AloudConcrete
• Demonstrate the units concretely (with different
colors)
• Verbally label the units for each set you are
putting together
• Match the demonstration to the strategy usage
the student is acquiring
• Verbally label the new unit
• Scaffold the placement of the manipulative
objects to the rationale counting progression (if
needed)
Your Turn
6 toy cars and 3 toy planes were on the table.
How many toys are on the table?
Model the following strategies concretely:
• Count all
• Count on
• Make 10 strategy
Your Turn
How would you model this problem with a
visual representation?
Why did you choose that representation?
Writing Equations
• 2 lollipops + 3 lollipops = 5 lollipops
• 2+3=5
• 5=2+3
• 6 cars + 3 airplanes = 9 toys
• 6+3=9
• 9=6+3
Concrete – Representational - Abstract
Dante had 7 Jolly Ranchers and 5 Snickers. How
many pieces of candy did he have?
7+5=?
7 + (3 + 2) = ?
(7 + 3) + 2 = ?
?=7+5
? = 7 + (3 + 2)
? = (7+3) + 5
Inverse Operations:
Add To / Take From and Put Together /
Take Apart
• Reversibility of actions:
– An Add To undoes a Take
From
– A composition (Put Together)
undoes a decomposition
(Take Apart)
Put Together/Take Apart
Both Addends Unknown
Grandma has 5 flowers. How many can she put
in her red vase and how many can she put in her
blue vase?
Put Together/Take Apart
Both Addends Unknown
• Allow students to explore various
combinations that make each number
• Facilitates level 2 and level 3 strategies
Teacher Model and Think AloudConcrete
• Demonstrate the total units concretely (with
different colors)
• Work systematically to show the different
combinations
• Verbally label the units
Your Turn
Mom has 7 cookies. How many can
she put in the red jar and how many
can she put in the blue jar?
Maintenance and Generalization
• Students need opportunities for cumulative
review during tier 2 instruction
• Students need opportunities to generalize
skills
Generalization
Problem type sorts:
Add To Result
Unknown
Put Together / Take
Apart Total Unknown
Take From Result
Unknown
Put Together / Take
Apart
Both Addends
Unknown
Questions
Denise.Schulz@dpi.nc.gov
Matt.Hoskins@dpi.nc.gov