Teaching Mathematics Meaningfully for Struggling Learners

1
Teaching Mathematics Meaningfully for Struggling
Learners

• LouAnn Lovin, Ph.D.
• Mathematics Education
• James Madison University

2
How would you respond to this student who
answered the following task as shown?

Which of the following helps you with 12 7 ?
a. 12 7 19 b. 2 5 7 c. 5 7
12 d. 4 5 9
Share your ideas with a neighbor.
3
Responding requires listening To make an
effort to hear something to pay attention
(Webster).

•  
• It seems simple enough
• Telephone Game, Grapevine, Pass It Down
• "We were given two ears but only one mouth,
  because listening is twice as hard as talking."  
  (Author unknown)

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Listen. . .

• What do you hear?

• There's a bathroom on the right
  

• Theres a bad moon on the rise

• The ants are my friends, is blowing in the wind

• The answer, my friends, is blowin in the wind

5

What do you hear?

• All of the other reindeer used to laugh and call
  him names
• OR
• Olive, the other reindeer, used to laugh and call
  him names
• Hold me closer, tiny dancer (Elton John)
• OR
• Hold me close, young Tony Danza
• On closed-captioned live television of a local
  news report a "grand
  parade" was captioned as a "Grandpa raid"
• http//www.phrases.org.uk/meanings/mondegreen.h
  tml

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Do you see what I see?

An old mans face or two lovers kissing?
Not everyone sees what you may see.
Cat or mouse?
An indians face or an Eskimo?
Young woman or old woman?
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What do you see?

Everyone does not necessarily hear/see/interpret
experiences the way you do.
www.news.com.au/dailytelegraph/story/0,22049,22535
838-5012895,00.html
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T Is four-eighths greater than or less than
four- fourths?J (thinking to himself)
Now thats a silly question. Four-eighths has
to be more because eight is more than four.
(He looks at the student, L, next to him who
has drawn the following picture.) Yup. Thats
what I was thinking.

Ball, D. L. (1992).  Magical hopes 
Manipulatives and the reform of mathematics
education (Adobe PDF).   American Educator,
16(2), 14-18, 46-47.
9
But because he knows he was supposed to show his
answer in terms of fraction bars, J lines up two
fraction bars and is surprised by the result

J (He wonders) Four fourths is more?T Four
fourths means the whole thing is shaded in.J
(Thinks) This is what I have in front of me. But
it doesnt quite make sense, because the pieces
of one bar are much bigger than the pieces of the
other one. So, whats wrong with Ls drawing?
Ball, D. L. (1992).  Magical hopes 
Manipulatives and the reform of mathematics
education (Adobe PDF). American Educator, 16(2),
14-18, 46-47.
10
T Which is more three thirds or five
fifths?J (Moves two fraction bars in front of
him and sees that both have all the pieces
shaded.)J (Thinks) Five fifths is more,
though, because there are more pieces.
This student is struggling to figure out what
he should pay attention to about the fraction
models is it the number of pieces that are
shaded? The size of the pieces that are shaded?
How much of the bar is shaded? The length of the
bar itself? Hes not seeing what the teacher
wants him to see.
Ball, D. L. (1992).  Magical hopes 
Manipulatives and the reform of mathematics
education (Adobe PDF).   American Educator,
16(2), 14-18, 46-47.
11
How would you respond to this student who
answered the following task as shown?

Which of the following helps you with 12 7 ?
a. 12 7 19 b. 2 5 7 c. 5 7
12 d. 4 5 9
12

• How you respond gives a glimpse into
• how you are listening
• what you are listening for
• and what you ignore
• It also sends a message to students about what
  is important in your classroom.

13
Possible Responsesdepends on how you are
listening

• No. You should have chosen (c) because it is part
  of the fact family.
• No. Choose another answer.
• What is 12 7? How did you get your answer?
• Explain how 2 5 7 helps you find the answer.
• Can you show me your reasoning using materials?

14
Childs explanation 12 - 7 ?

• L What is 12-7?
• C 5
• L How did you get that?
• C Well, I know 12 is 2 away from 10, so I broke
  7 into a 2 and a 5. Then I took away 2 from both
  12 and 7 so that I had a 10 and a 5. I know what
  10 - 5 is. Its 5.
• L Why did you choose (b)?
• C Because its 2 5 7 and I used those
  numbers to find the answer.

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Different kinds of listening

• Evaluative
• Response seeking
• Listening for particular responses
• Set learning trajectory
• Interpretive
• Information seeking
• Making sense of students sense-making
• Listening for particular responses
• Set learning trajectory
• Hermeneutic
• Moving with the students
• Mathematical ideas are locations for exploration
• Student contributions essentially direct the
  learning trajectory of the class

Davis, B. (1997). Listening for differences An
evolving conception of mathematics teaching.
Journal for Research in Mathematics Education
28(3), 355-376. Reston, VA NCTM.
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Sowhat does listening and seeing have to do
with struggling learners?
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Struggling Learners

• Oftentimes are interpreting or focusing on things
  that are different from the teacher.
• Too often only experience a remediation
  mathematics focused on procedures and a teacher
  who is only listening evaluatively.
• End up with Swiss Cheese for knowledge and
  understanding.

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Response to Intervention
Tier 3 Individualized intervention(s) of
increased intensity for students
who show minimal response to
secondary prevention

Tier 2 Evidence-based intervention(s) of
moderate intensity that addresses the
learning or behavioral challenges of
most at-risk students
Tier 1 High quality core instruction that meets
the needs of most students
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Average Effect Sizes for Instructional Variables
for Special Education Students Other
Low-Achieving Students
Instructional Strategy Effect Size for Special Education Students Effect Size for Low-Achieving Students
Visual and graphic depictions of problems 0.50 Moderate N/A
Systemic and explicit instruction 1.19 Large 0.58 Moderate to Large
Student think-alouds 0.98 Large N/A
Use of structured peer-assisted learning activities involved heterogeneous ability groupings 0.42 Moderate 0.62 Large
Formative assessment data provided to teachers 0.32 Small to Moderate 0.51 Moderate
Formative assessment data provided directly to students 0.33 Small to Moderate 0.57 Moderate to Large
NCTM Research Brief
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Evidence-Based Practices

• Visual and graphic depictions of problems
• Systemic and explicit instruction
• Student think alouds

While these practices were investigated with
struggling learners, these can be good practices
for all students (with some modifications).
So well consider these practices in a general
education setting (Tier 1) as well as discuss
issues of which to be aware with struggling
learners (Tier 1-3).
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Solve the Problem

• A big dog weighs five times as much as a little
  dog. The little dog weighs 2/3 as much as a
  medium-sized dog. The medium-sized dog weighs 9
  pounds more than the little dog. How much does
  the big dog weigh?

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1. Visual and Graphic Depictions

• It was not whether teachers used visual/graphic
  depictions, it was how they were using them.
• Graphic depictions of multiple problems
• Students using their own graphic depictions and
  receiving feedback/guidance from the teacher
• Discussions about why particular representations
  might be more beneficial to help think through a
  given problem or communicate your ideas.
• Use of manipulatives and visuals with older
  students (middle and high school)

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A big dog weighs five times as much as a little
dog. The little dog weighs 2/3 as much as a
medium-sized dog. The medium-sized dog weighs 9
pounds more than the little dog. How much does
the big dog weigh?

• Let x weight of medium dog.
• Then 2/3 x weight of small dog.
• Then 5(2/3 x) weight of big dog.
• x 9 2/3 x (med 9 small)
• 1/3 x 9
• x 27 pounds
• 2/3 x 18 pounds
• 5(2/3 x) 5(18) 90 pounds

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A big dog weighs five times as much as a little
dog. The little dog weighs 2/3 as much as a
medium-sized dog. The medium-sized dog weighs 9
pounds more than the little dog. How much does
the big dog weigh?
weight of medium dog
9
9
9
weight of small dog
9
9
18
18
18
18
18
weight of big dog
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A big dog weighs five times as much as a little
dog. The little dog weighs 2/3 as much as a
medium-sized dog. The medium-sized dog weighs 9
pounds more than the little dog. How much does
the big dog weigh?
x weight of medium dog
9
9
9
x
2/3 x weight of small dog
9
9
2/3 x
18
18
18
18
18
5 (2/3 x)
5(2/3 x) weight of big dog
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Completing the Squarex2 2x 4

x2
2x
4

2
x
x
2
x
2
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x2 2x 4

x2
2x
4

2
x
x
2
x
2
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x2 2x 4

x
1

x
x2
4

1
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x2 2x (½(2))2 4 (½(2))2

x
1

x
x2
4

1
30
x2 2x (½(2))2 4 (½(2))2

x
1
1
1
x
x2
4


1
x2 2x 1 4 1 (x 1) 2 5
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Visual and Graphic Depictions

• Singapore Math (Quantitative Analysis)
• 1. Word Problems
• Introduce procedures and concepts using word
  problems (e.g., long division multiplication and
  division of fractions).
• Makes learning more concrete by presenting
  abstract ideas in a familiar context.
• AVOIDs the sole reliance on key words.
• 2. (Particular) Visual models
• Help children to get past the words by visualizing
   and  illustrating word problems
  with simple diagrams.
• Scaffolding 

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NCTM Process Standards
Communication
Representations
Reasoning
Connections
Problem Solving
Principles and Standards of School Mathematics,
NCTM, 2000.
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1. Visual and Graphic Depictions

• Students using their own graphic depictions and
  receiving feedback/guidance from the teacher
  (During class and through Mathematical Write-Ups)
• Considering the advantages and disadvantages of
  different visuals.

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Different visual depictions of problem solutions
for the same problem
Sara
Kevin
Natalie
Katie
2
Kevin
Sara
Natalie
Katie
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2. Systemic and Explicit Instruction

• What does Explicit Instruction mean to you?

Share your ideas with a neighbor.
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2. Systemic and Explicit Instruction

• Instruction that involves a teacher demonstrating
  a specific plan (strategy) for solving given
  problem types.
• Students use this plan to think their way through
  to a solution.
• Emphasis is placed on providing highly explicit
  models of steps and procedures or questions to
  ask in solving problems.
• NOTE The majority of studies dealt with
  procedural knowledge.

How is this different from traditional
instruction?
Gersten, R. Clarke, B. (2007). Research Brief
Effective Strategies for Teaching Students with
Difficulties in Mathematics. NCTM Reston, VA.
37
My Take on Explicit Instruction

• It is NOT someone rotely teaching steps to a
  procedure.
• Someones (peer or teacher) THINKING about a
  problem should be made as explicit (clear) as
  possible (using visuals, color coding,
  think-alouds) to help students "see" what another
  person is thinking.
• Sharing strategies and ideas is paramount.
• Teacher looks for significant ideas to highlight
  (think back to the fraction bar example).

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Systemic and Explicit Instruction

• Visual Cuing
• Color Coding
• Highlighting significant ideas in students work

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Visual Cuing
Grade 6 Math SOL 6.10
Area All Over
Perimeter
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Systemic and Explicit Instruction

• Word of Caution
• In Tier 2 and 3 (and sometimes in Tier 1) a
  teacher might feel the need or be more
  comfortable simply showing students a particular
  strategy to use and practice (oftentimes the most
  sophisticated way of thinking about the idea).
• There are different developmental levels students
  move through for given mathematical ideas. So
  which strategy will you demonstrate?
• Levels of CGI Strategies
• Obviously, pre-assessment is key here. How are
  you using pre-assessment to inform your
  instruction?

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A Common Approach for Struggling Learners

• Randomly combining numbers without trying to make
  sense of the problem.

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43

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Key Words

• This strategy is useful but limited because key 
  words don't help students understand the 
  problem situation (i.e. what is happening 
  in the problem). 
• Key words can also be misleading because the 
  same word may mean different things in 
  different situations. 
• There are 7 boys and 21 girls in a class. How many
   
• more girls than boys are there? 
• There are 21 girls in a class. There are 3 times a
  s many girls as boys. How many boys are in the cla
  ss?

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• Real world problems do not have key words!

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3. Student Think Alouds

• Encourage students to verbalize their thinking
  (talking, writing, or drawing) the steps they
  used in solving a problem.
• Addresses the impulsiveness of randomly combining
  numbers instead of implementing a solution
  strategy step by step.

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My Kids Can 2nd grade video (15 mins)

• Notice the teachers comments and questions to
  the students responses.
• How is he listening?
• What ideas is he trying to make explicit to the
  students?
• Notice the opportunities students have to share
  their reasoning.

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3. Student Think Alouds

• Students sharing their ideas
• Helps struggling learners realize that others are
  using strategies
• Helps students clarify their reasoning
• Helps students see there are multiple ways to
  solve a task
• Helps the teacher assess student understanding
• Helps send the message that mathematics is more
  than right or wrong answers.

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Take Aways

• Evidence-Based Practices
• Visual and graphic depictions of problems
• Systemic and explicit instruction
• Student think alouds

Investigated with struggling learners, but can be
good practices for all students with some
modifications (think procedural and conceptual
knowledge).BUT
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Listening only for answers
interpreting explicit as simply telling

Take Aways
problems in reaching learners,
especially struggling learners.
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What are you listening for. . .
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References

• Allsopp, D., Kyger, M., Lovin, L. (2007).
  Teaching Mathematics Meaningfully Solutions for
  Reaching Struggling Learners. Brookes Baltimore,
  MD.
• Carpenter, Fennema, Franke, Levi, Empson. (1999).
  Childrens Mathematics Cognitively Guided
  Instruction. Heinemann Portsmouth, NH.
• Gersten, R. Clarke, B. (2007). Research Brief
  Effective Strategies for Teaching Students with
  Difficulties in Mathematics. NCTM Reston, VA.
• Ministry of Education Singapore. (2009). The
  Singapore Model Method. Panpac Education
  Singapore.
• NCTM (2000). Principles and Standards of School
  Mathematics. NCTM Reston, VA.
• Parrish, S. (2010). Number Talks Helping
  Children Build Mental Math and Computation
  Strategies. Math Solutions Sausalito, CA.
• Storeygard, J. (2009). My Kids Can Making Math
  Accessible to All Learners. Heinemann
  Portsmouth, NH.

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Cognitively Guided InstructionStrategies

• Direct Modeling Strategies
• Counting Strategies
• Derived Number Facts
• Known Number Facts (as in recall)

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