Download Helping Struggling Readers with Mathematical Reasoning

Helping Struggling Learners
with Mathematical Reasoning
David J. Chard
Southern Methodist University
Simmons School of Education and Human Development
Purposes of the Presentation
• Discuss the ingredients needed to create a
system of mathematics education that will
ensure greater achievement for all students.
• Consider the needs of teachers and students
to improve mathematics achievement.
• Review evidence to identify and provide
examples of instructional practices that can be
incorporated into core instruction and
interventions
Powerful Core Mathematics Instruction
System/
Physical
Environment
Standards
and
Expectations
Teacher
Knowledge
and
Practice
Instructional
Tools and
Materials
Enhanced
Student
Learning and
Achievement
Connections for Effective
Mathematics Teaching
Teacher
Mathematics
For Teaching
Tools, Practices,
Programs
Fundamental
Mathematics
Student Needs
http://www.NCTQ.org
Understanding Fundamentals of
Mathematics
• Historical development
• Number systems and relationships
The Real Numbers
Defines Division
Defines Addition,
Subtraction, and
Multiplication
Understanding Fundamentals of
Mathematics
•
•
•
•
Historical development
Number systems and relationships
Key mathematical principles
Appropriate models of mathematical concepts
and dispelling common misconceptions
Modeling Fractions
Set Models
Number Line
Area Models
Fraction Equivalence
Rates
Part-Whole Relations
3
4
Quotient
3
4
of
Operations
Understanding Fundamentals of
Mathematics
•
•
•
•
Study historical development of mathematics
Understand number systems and relationships
Know key mathematical principles
Prepare appropriate models of mathematical
concepts and dispel common misconceptions
• Understand that common algorithms are
derived from formulas
Demystifying Mixed Numbers
Definition of fraction addition
2
3
5
Understanding Fundamentals of
Mathematics
•
•
•
•
Study historical development of mathematics
Understand number systems and relationships
Know key mathematical principles
Prepare appropriate models of mathematical
concepts and dispel common misconceptions
• Understand that common algorithms are
derived from formulas
• Develop a positive disposition for
mathematics and problem solving
Connections for Effective
Mathematics Teaching
Teacher
Mathematics
For Teaching
Tools, Practices,
Programs
Fundamental
Mathematics
Student Needs
Common Difficulty Areas for Diverse Learners
(Student Needs) (Baker, Simmons, & Kame’enui, 1995)
Memory and Conceptual
Difficulties
Background Knowledge
Deficits
Linguistic and Vocabulary
Difficulties
Strategy Knowledge
and Use
Addressing Diverse Learners Through
Effective Core Instruction
Thoroughly develop concepts,
principles, and strategies.
Memory and
Conceptual
Difficulties
Gradually develop knowledge
and skills while ensuring student
success.
Include non-examples to
teach students to focus on
relevant features.
Include a planful system of
review.
Presenting Mathematical Ideas Conceptually
Definition
Model
Two or more fractions
that represent the
same point on the
number line
Equivalent
fractions
2/3 ~ 4/6
2/5 ~ 4/5
8/4 ~ 2/1
3/2 ~ 3/4
Examples
Non-Examples
Sequenced Examples
Combining Like Terms
A. 2X + 4X =
Identify like terms.
6X
Combine coefficients
2+4=6
B. 3c – 2d + 2c + 5
Identify like terms.
5c – 2d + 5
Combine coefficients
3+2=5
C. 2k + 4m – 2m + k
Identify like terms.
3k + 2m
Combine coefficients
2+1=3
4–2=2
D. 7f + 2g – 2
No like terms to combine.
Addressing Diverse Learners Through
Effective Core Instruction
Identify and pre-teach
prerequisite knowledge.
Background
Knowledge
Deficits
Assess background
knowledge.
Differentiate practice and
scaffolding.
Build From Previous Knowledge
Numerical Expression
Algebraic Expression
7+3
n+3
variable
constant
Here is how you read algebraic expressions
sum
n+3
product 3 x n or 3n
“n plus 3”
“3 times n”
difference
quotient
n–3
“n less 3”
n÷3 “n divided by 3”
Addressing Diverse Learners Through
Effective Core Instruction
Define and use mathematical
symbols with precision and in
multiple contexts.
Linguistic and
Vocabulary
Difficulties
Describe and develop
vocabulary deliberately and
thoughtfully.
Model and encourage the use
of mathematical vocabulary
in classroom discourse.
Provide students opportunities
to discuss mathematics and to
receive feedback on their
discussions.
A Plan for Vocabulary in
Mathematics
1.
Assess students’ current knowledge.
2.
Teach new vocabulary directly before and during reading of
domain specific texts.
3.
Focus on a small number of critical words.
4.
Provide multiple exposures (e.g., conversation, texts,
graphic organizers).
5.
Engage students in opportunities to practice using new
vocabulary in meaningful contexts.
(Baker, Gersten, & Marks, 1998; Bauman, Kame’enui, & Ash, 2003;
Beck & McKeown, 1999; Nagy & Anderson, 1991; Templeton, 1997)
Sample Text
Like terms can be combined because they have the same variable
raised to the same power. They may have different coefficients. When
you combine like terms, you simplify the expression but you do not
change the value of the expression. Equivalent expressions have the
same value for all values of the variables.
Key Vocabulary
Like Terms
Coefficients
Simplify
Equivalent expressions
Discuss
1. What is the first step in simplifying the expression
3 + 2(d+3f) - 2f?
2. How many different sets of like terms are in the expression in (1) above?
3. Explain why 3a + 3b + 3c is already simplified.
The purpose of teaching students key mathematical
vocabulary is to anchor their understanding in words
that they can then use to communicate mathematically.
Foster Students’ Ability to Translate
Translate the expressions
Mathematics Expression
3x + 2
Sara has x cds;
Daisy has 2x + 3 cds
English Expression
Two more than the product of 3 and x
Sara had some cds; Daisy had
three more than twice as many
as Sara
Mathematics Expression
English Expression
2(3x-1)
49 less than the value of some
number is 65. What is the
number?
Mathematics Expression
English Expression
1/2b = 12
If you divide a number by 5, the
result is 30. What is the number?
Addressing Diverse Learners Through
Effective Core Instruction
Model problem-solving
strategies through thinkalouds.
Strategy Knowledge
and Use
Teach the “how”, “why”, and
“when” of strategy application.
Have students verbally
rehearse the steps to solving
problems.
Provide interactive problemsolving opportunities.
Verbally rehearse the steps.
RtI: Multi-Tiered Model
RtI:Multi-Tiered Model
Focus
Program
Interventionist
Setting
Grouping
Time
Assessment
A Closer Look at Tier II
A Closer Look at Tier II
Focus
Program
Interventionist
Setting
Grouping
Time
Assessment
Research and Practice Relationship
Research – practice = irrelevant
Practice – research = irresponsible
Research + practice = relevant and responsible
Dr. Kame’enui, 2009
Explicit and Systematic Instruction
Teacher provides clear models
Students receive extensive and scaffolded practice
and review
Teacher and students verbalize and think aloud
Teachers provides corrective feedback
Elementary and Middle
School Settings
• Level of evidence: Strong
• IES Practice Guide
Recommendation 3
Secondary School Settings
• Level of evidence: Promising
Learning Outcomes
• Research findings:
– Improved proficiency in solving word problems
– Improved procedural fluency
• Possible outcomes:
– Connection between conceptual and procedural
understanding
– Generalize concepts and procedures
– Ability to communicate problem solving and
reasoning
Visual Representations
Model mathematical concepts using visual
representations
Relationship is explicitly taught
Middle School Settings
• Level of evidence:
Moderate
• IES Practice Guide
Recommendation 5
Secondary School Settings
• Level of evidence: Promising
Considerations for Using
Visual Representations
• Explicitly link the concrete model and visual
representation with the abstract mathematics
• Use consistent language across
representations
• Provide ample practice opportunities
• Select examples and non-examples carefully
Sequence and Range of Examples
Sequence of examples is systematic
Variation in range of examples
Example: Concrete-RepresentationalAbstract
Middle School Settings
• Level of evidence:
Moderate
• Effect size: 0.82 (p<.001)
Secondary School Settings
• Level of evidence: Possible
⅞−⅕
⅓÷⅘
¾ = 0.75
0.54 × 0.2
21 ÷
⅖
Abstract
Representation
(Formal)
Visual
Representations
(Pre-formal)
Concrete Models
(Informal)
Webb, Boswinkel, Dekker, 2008
Learning Outcomes
• Research findings:
– Improved achievement in general mathematics,
pre-algebra concepts, word problems, operations
• Possible outcomes:
– Understand meaning of abstract symbols
– Develop strategies for translating word problems
into abstract numerical statements
– Scaffold students’ conceptual understanding
Developing Understanding of Fractions
Concrete
Visual Representations
Best Evidence Synthesis:
Effective Programs in Middle and High School
Mathematics
Slavin, Lake, & Groff (2009)
• Best evidence syntheses are similar to metaanalyses, pooling effects across studies to
determine the contribution of a particular
approach.
• Very similar to the What Works Clearinghouse
model.
Findings
• Programs and approaches effective with one
subgroup, appear to be effective with all subgroups
• There is a lack of evidence that particular curricular
materials are better than others (e.g., NSF-supported
vs. non-NSF supported)
• Approaches that produce significantly positive effects
are those that fundamentally change what students
do every day in the classroom
• What new information have
you learned?
• What remains confusing or
unanswered?
• How might you apply what
you’ve learned?
What I Believe We Must Do
• Take responsibility for our children’s mathematics
development
• Determine what we need to learn and invest our
time and energy to make it happen
• Focus our attention on the whole picture;
improvement in one aspect is insufficient
• Sustain our effort to improve mathematics
instruction for all children
• Develop sophisticated measurement systems to track
progress and make difficult resource allocations to
ensure success
Thank You