Linking Mathematics Achievement to Successful Mathematics Learning throughout Elementary School: Cri

1
Linking Mathematics Achievement to
Successful Mathematics Learning throughout
Elementary School Critical Big Ideas and Their
Instructional Applications Ben Clarke,
Ph.DPacific Institutes for ResearchFebruary 23,
2005
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Contact Information

• Email
• clarkeb_at_uoregon.edu
• Phone
• [phone]
• Special thanks to David Chard, Scott Baker,
  Russell Gersten, and Bethel School District

3
Warm up Two machines one job

• Rons Recycle Shop was started when Ron bought a
  used paper-shredding machine. Business was good,
  so Ron bought a new shredding machine. The old
  machine could shred a truckload of paper in 4
  hours. The new machine could shred the same
  truckload in only 2 hours. How long will it take
  to shred a truckload of paper if Ron runs both
  shredders at the same time?

4
A primary goal of schools is the development of
students with skills in mathematics

• Mathematics is a language that is used to express
  relations between and among objects, events, and
  times. The language of mathematics employs a set
  of symbols and rules to express these relations.

(Howell, Fox, Morehead, 1993)
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Numbers are abstractions

• To criticize mathematics for its abstraction is
  to miss the point entirely. Abstraction is what
  makes mathematics work. If you concentrate too
  closely on too limited an application of a
  mathematical idea, you rob the mathematician of
  his or her most important tools analogy,
  generality, and simplicity (Stewart, 1989, p.
  291)
• The difficulty in teaching math is to make an
  abstract idea concrete but not to make the
  concrete interpretation the only understanding
  the child has (i.e. generalization must be
  incorporated).

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The Number 7

• Could be used to describe
• Time
• Temperature
• Length
• Count/Quantity
• Position
• Versatility makes number fundamental to how we
  interact with the world

7
Mathematical knowledge is fundamental to function
in society

• For people to participate fully in society, they
  must know basic mathematics. Citizens who cannot
  reason mathematically are cut off from whole
  realms of human endeavor. Innumeracy deprives
  them not only of opportunity but also of
  competence in everyday tasks. (Adding it Up,
  2001)

8
Proficiency in mathematics is a vital skill in
todays changing global economy

• Many fields with the greatest rate of growth will
  require workers skilled in mathematics.
  (Bureau of Labor Statistics,1997)
• Companies place a premium on basic mathematics
  skill even in jobs not typically associated with
  mathematics.
• Individuals who are proficient in mathematics
  earn 38 more than individuals who are not.
  (Riley, 1997)

9
Despite the efforts of educators many students
are not developing basic proficiency in
mathematics

• Only 21 of fourth grade students were classified
  as at or above proficiency in mathematics, while
  36 were classified as below basic. This pattern
  was repeated for 8th and 12th grade (NAEP, 1996).
• According to the TIMS (1998), US students perform
  poorly compared to students in other countries.
  United States 12th graders ranked 19th out 21
  countries.
• The result Students lack both the skills and
  desire to do well in mathematics (MLSC, 2001)

10
Achievement stability over time

• The inability to identify mathematics problems
  early and use formative evaluation is problematic
  given the stability of academic performance.
• In reading, the probability of a poor reader in
  Grade 1 being a poor reader in Grade 4 is .88
  (Juel, 1988).
• The stability of reading achievement over time
  has led to the development of DIBELS.

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Trajectories The Predictions

• Students on a poor reading trajectory are at
  risk for poor academic and behavioral outcomes in
  school and beyond.
• Students who start out on the right track tend to
  stay on it.

(Good, Simmons, Smith, 1998)
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Developmental math research

• Acquisition of early mathematics serves as the
  foundation for later math acquisitions (Ginsburg
  Allardice, 1984)
• Success or failure in early mathematics can
  fundamentally alter a mathematics education
  (Jordan, 1985)

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Discussion Point Trajectories

• Do math trajectories and reading trajectories
  develop in the same way?
• How could they be similar?
• How could they be different?
• Are they the same for different types of learners
  (e.g. at-risk)?

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Discussion Point Number Sense

• What is number sense?
• What does number sense look like for the
  grade/students you work with?
• How does number sense change over time and what
  differentiates those with and without number
  sense over time?

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The Ghost in the MachineNumber Sense

• Number sense is difficult to define but easy to
  recognize
• (Case, 1998)
• Nonetheless he defined it!

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Case (1998) Definition

• Fluent, accurate estimation and judgment of
  magnitude comparisons.
• Flexibility when mentally computing.
• Ability to recognize unreasonable results.
• Ability to move among different representations
  and to use the most appropriate representation.

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Cases Definition (cont.)

• Regarding fluent estimation and judgment
• of magnitude (i.e. rate and accuracy).
• Recent empirical support for this insight
• Landerl, Bevan, Butterworth
• (2004), 3rd grade
• Passolunghi Siegel (2004),
• 5th grade

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Number Sense

• A key aspect of various definitions of number
  sense is flexibility.
• a childs fluidity and flexibility with numbers,
  the sense of what numbers mean, and an ability to
  perform mental mathematics and to look at the
  world and make comparisons
• Students with number sense can use numbers in
  multiple contexts in multiple ways to make
  multiple mathematics decisions.

(Gerston Chard, 1999)
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Math LD

• 5 to 8 percent of students
• Basic numerical competencies are intact but
  delayed
• Number id
• Magnitude comparison
• Difficulty in fact retrieval
• Proposed as a basis for RTI and LD diagnosis

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Math LD (cont.)

• Working memory deficits hypothesized to underlie
  fact retrieval difficulty
• Students use less efficient strategies in
  solving math problems due to memory deficits
• Procedural deficits often combine with conceptual
  misunderstanding to make solving more complex
  problems difficult.

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Initial Comments about Mathematics Research
The knowledge base on documented effective
instructional practices in mathematics
is less developed than reading.
Mathematics instruction has been a concern to
U.S. educators since the 1950s, however,
research has lagged behind that of
reading.
Efforts to study mathematics and mathematics
disabilities has enjoyed increased interests
recently.
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Purpose

• To analyze findings from experimental research
  that was conducted in school settings to improve
  mathematics achievement for students with
  learning disabilities.

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Identifying High Quality Instructional Research
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Method

• Included only studies using experimental or
  quasi-experimental group designs.
• Included only studies with LD or LD/ADHD samples
  OR studies where LD was analyzed separately.
• Only 26 studies met the criteria in a 20-year
  period (Through 1998).

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RESULTSFeedback to Teachers on Student
Performance

• Seems much more effective for special educators
  than general educators, though there is less
  research for general educators.
• May be that general education curriculum is often
  too hard.

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Feedback to Teachers on Student Performance
(cont.)

• 3. Always better to provide data and
  suggestions rather than only data profiles (e.g.,
  textbook pages, examples, packets, ideas on
  alternate strategies).

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Feedback to Students on Their Math Performance

• Just telling students they are right or wrong
  without follow-up strategy is ineffective (2
  studies).
• Item-by-item feedback had a small effect (1
  study).
• Feedback on effort expended while students do
  hard work (e.g., I notice how hard you are
  working on this mathematics) has a moderate
  effect on student performance.

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Goal Setting

• Studies that have used goal setting as an
  independent variable, however, show effects that
  have not been promising.
• Fear of failure?
• Requires too much organizational skill?

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Peer Assisted Learning

• Largest effects for well-trained, older students
  providing mathematics instruction to younger
  students
• Modest effect sizes (.12 and .29) were documented
  for LD kids in PALS studies. These were
  implemented by a wide range of elementary
  teachers (general ed) with peer tutors who didnt
  receive any specialized training (More recent
  PALS data not included)

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Curriculum and Instruction

• Explicit teacher modeling, often accompanied by
  student verbal rehearsal of steps and
  applications
• -moderately large effects
• Teaching students how to use visual
  representations for problem solving
• -moderate effects

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Key Aspects of Curriculum Findings

• The research, to date, shows that these
  techniques work whether students do a lot of
  independent generation of the think alouds or the
  graphics or whether students are explicitly
  taught specific strategies

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Overview of Findings

• Teacher modeling and student verbal rehearsal
  remains phenomenally promising and tends to be
  effective.
• Feedback on effort is underutilized and the
  effects are underestimated.
• Cross-age tutoring seems to hold a lot of promise
  as long as tutors are well trained.
• Teaching students how to use visuals to solve
  problems is beneficial.
• Suggesting multiple representations would be
  good.

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Big Ideas in Math Instruction

• Math instruction should build competency within
  and across different strands of mathematics
  proficiency
• Math instruction should link informal
  understanding to formal mathematics
• Math instruction should be based on effective
  instructional practices research

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Big Idea Five Strands of Mathematical Proficiency

• 1. Conceptual Understanding-comprehension of
  mathematical concepts, operation, and relations
• 2. Procedural Fluency-skill in carrying out
  procedures flexibly, accurately, efficiently, and
  appropriately
• 3. Strategic Competence-ability to formulate,
  represent, and solve mathematical problems

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Five Strands (cont.)

• 4. Adaptive Reasoning-capacity for logical
  thought, reflection, explanation, and
  justification
• 5. Productive Disposition-habitual inclination to
  see mathematics as sensible, useful, and
  worthwhile, coupled with a belief in diligence
  and one's own efficacy.

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Big Idea Informal to Formal Mathematics

• Early math concepts are linked to informal
  knowledge that a student brings to school
  (Jordan, 1995)
• Linking informal to formal math knowledge has
  been a persistent theme in the mathematics
  literature (Baroody, 1987)

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Counting

• Sequence words w/out reference to objects
• 1) through 20 is unstructured learned through
  rote memorization
• 2) students learn 1-9 repeated structure
• 3) students learn decade transitions

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Counting (cont.)

• Counting occurs when sequences words are assigned
  to objects on a one to one basis
• Counting first step in making quantitative
  judgments about the world exact

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Counting (cont.)

• 5 Principles of Counting
• One to one correspondence
• Stable order principle
• Cardinal principle (critical)
• Item indifference
• Order indifference
• (Gelman Gallistel, 1978)

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Cardinality

• Developed around the age of 4 (Ginsburg Russel,
  1981)
• All or nothing phenomena(Permangent, 1982)
• Can be taught and focus children on seeing
  individual items in terms or being part of a
  larger unit (Fuson Hall, 1983)

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From counting to addition

• Addition makes counting abstract
• Addition is counting sets
• 2 apples and 3 apples

42
The link to addition

• Count All starting with First addend (CAF)
• Count All starting with Larger addend (CAL)
• Count On from First addend (COF)
• Count on from Larger addend (COL)

43
Development of early addition (cont.)

• Students first use CAF and COF supporting a
  uniary view of addition (e.g. changing one
  number)
• CAL and COL supports a binary (I.e. combining two
  number) view of addition
• Based on principle of commutativity
• Students who understand communtativity can use
  the COL strategy

44
Addition Strategies

• COL strategy has been termed the Min strategy
  because it requires the minimal amount of
  counting steps to solve a problem
• Recognized as the most cognitively efficient

45
Addition (cont.)

• Some problems were solved quicker than expected
• Based on patterns such as doubles, tens
• Indicate development of number sense
• (Groen Parkman, 1972)
• Siegler (1982) hypothesized that use of the min
  effect was the critical variable in 1st grade
  math and failure to do so was predictive of later
  failure in mathematics

46
Big Idea Effective Instruction

• Five key components of effective instruction are
• Big Ideas
• Conspicous strategies
• Review/Reteaching
• Scaffolding
• Integration

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Mathematical concepts, operations, and strategies
that

• form the basis for further mathematical
  learning.

• map to the content standards outlined by your
  state.

• are sufficiently powerful to allow for broad
  application.

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Mathematics Big Ideas

• Place Value Place a number holds gives
  information about its value
• Expanded Notation A number can be reduced to its
  parts (e.g. 432 is 400 30 and 2)
• Commutative property a b b a
• Equivalence Quantity to the left and right of
  equal sign are equivalent 32 15 47
• Rate of composition/decomposition Rate is base
  10 system is 10.

(Kamenui et al, 1998)
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Conspicuous Strategies
Expert actions for problem solving that are made
overt through teacher and peer modeling.
In selecting exemplars consider teaching

• the general case for which the strategy works.

• both how and when to apply a strategy.

• when the strategy doesnt work.

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Time
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Instructional Scaffolding includes
Sequencing instruction to avoid confusion of
similar concepts.
Carefully selecting and sequencing of examples.
Pre-teaching of prerequisite knowledge.
Ensuring mathematical proficiency when necessary.
Providing specific feedback on students efforts.
Offering ample opportunities for students to
discuss their approaches to problem solving.
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Integration
it is not necessary that students master place
value before they learn a multi-digit algorithm
the two can be developed in tandem. --(Mathemati
cs Learning Study Committee, 2002)
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Review/Reteaching
Review must be

• sufficient to enable a student to perform the
  task
• without hesitation,

• distributed over time,

• cumulative with less complex information
  integrated
• into more complex tasks, and

• varied to illustrate the wide application of a
  students
• understanding of the information.

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Discussion Point Effective Instructional
Principles

• How are effective instructional principles likely
  to vary by student skill level
• Big Ideas
• Conspicuous Strategies
• Scaffolding
• Integration
• Review/Reteach

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Big Idea - Addition
Plan and design instruction that

• Develops student understanding from concrete to
  conceptual,

• Applies the research in effective math
  instruction, and

• Scaffolds support from teacher ? peer ?
  independent application.

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Sequencing Skills and Strategies
Concrete/ conceptual
Adding w/ manipulatives/fingers Adding w/
semi-concrete objects (lines or dots) Adding
using a number line Min strategy Missing addend
addition Addition number family facts Mental
addition (1, 2, 0) Addition fact memorization
Abstract
Abstract
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Sequence of Instruction
1.
2.
4.
3.
58
1.
2.
4.
3.
1. Teach prerequisite skills thoroughly.
6 3 ?
What are the prerequisite skills students need to
master to introduce adding single digit numbers?
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1.
2.
4.
3.
2. Teach easier skills and strategies before
more difficult ones.
Vertical, horizontal, or mix?
Sums to 5, 10, or 18?
Adding any number 0-9 or any number 1-9?
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1.
2.
4.
3.
3. Introduce strategies one at a time until
mastered. Separate strategies that are
potentially confusing.
As you teach students to add, when should you
introduce subtraction?
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1.
2.
4.
3.
3.
62
1.
2.
4.
3.
4. Introduce new skills, strategies, and
applications over a period of time through a
series of lessons beginning with lots of
teacher modeling, guided practice,
integration, independent practice, and review.
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Scaffold the Instruction
Time
64
Lesson Planning Addition with Manipulatives
Scaffolding
Teacher Monitored Independent
Independent (no teacher monitoring)
Guide Strategy
Strategy Integration
Model
Day 1 2 problems
Day 3 4 problems
Day 5 6 problems
Day 7 8 problems
Day 9 until accurate
Until fluent
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Selection of examples
Selection of Examples
Which problems would be appropriate for
introducing students to addition?
What misrules might students make?
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Selection of examples
2 5
6 2
1. Choose examples with single digit addends
and sums up to 10.
3 6 7 0
2. Write problems with a mix of larger and
smaller addends first.
3. Start with horizontal alignment, then
introduce vertical alignment, then mix.
4. Introduce with addends 1-9, then introduce
addends of 0.
67
Introduction to the Concept of Addition
68
Addition of Semi-concrete Representational Models
5 3 ?
69
The Min Strategy
5 3 ?
8
70
Missing Addend Addition
4 ? 6
71
Mental Math
5 1
1 2 3 4 5 6 7 8 9 10
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Mental Math
5 1
1 2 3 4 5 6 7 8 9 10
73
Mental Math
5 2
1 2 3 4 5 6 7 8 9 10
74
Mental Math
5 2
1 2 3 4 5 6 7 8 9 10
75
Number Families
4 3
7
76
Fact Memorization
4 3
1 8
5 2
6 0
77
Lesson Planning Addition with Manipulatives
Scaffolding
Teacher Monitored Independent
Independent (no teacher monitoring)
Guide Strategy
Strategy Integration
Model
Day 1 2 problems
Day 3 4 problems
Day 5 6 problems
Day 7 8 problems
Day 9 until accurate
Until fluent
78
Steps in Building Computation Fluency (Van de
Walle 2004)

• Direct Modeling
• Counts by Ones ---- Ten Frames
• Invented Strategies
• Written Records
• Mental Math
• Traditional Algorithms
• Guided development

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Invented Strategies/Traditional Algorithms

• Van de Walle states
• Invented strategies are number orientated rather
  than digit orientated
• 45 32 focus is on 40 30 rather than 4 3
• Emphasis on place value
• Invented Strategies are left handed vs. right
  handed
• 26x47 start with 20x40

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Invented Strategies (cont.)

• Invented Strategies are flexible rather than
  rigid
• 7000 -25 traditional algorithm requires complex
  steps

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Exercise

• Work with a partner to come up with three ways
  students could solve
• 46 38 ?

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Invented Strategies Addition examples 4638 ?

• Add Tens, Add Ones, Combine
• 40 and 30 is 70
• 6 and 8 is 14
• 70 and 14 is 84
• Add on Tens, then Add Ones
• 40 and 36 is 76
• Then add on 8, 76 and 4 is 80 and another 4 is 84

83
Invented Strategies Addition examples 4638 ?

• Add Tens, Add Ones, Combine
• 40 and 30 is 70
• 6 and 8 is 14
• 70 and 14 is 84
• Add on Tens, then Add Ones
• 40 and 36 is 76
• Then add on 8, 76 and 4 is 80 and another 4 is 84

84
Invented Strategies Addition examples 4638 ?

• Move some to make tens
• 2 from 46 put with 38 to make 40
• You have 44 and 40 more is 84
• Use a nice number and compensate
• 46 and 40 is 86
• I used 2 extra so 84

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Discussion Point Invented Strategies

• With a partner discuss
• What works about invented strategies
• How would you incorporate invented strategies
  into your classroom
• Would invented strategies work equally well for
  all students.

86
Big Idea - Functions
Plan and design instruction that

• Develops student understanding from concrete to
  conceptual,

• Applies the research in effective math
  instruction, and

• Scaffolds support from teacher ? peer ?
  independent application.

87
Sequencing Skills and Strategies
Concrete/ conceptual
Identify examples of simple functions (temperature
conversions, rate-kph) Distinguishing linear
from non-linear functions in graphs Determine
and graph ordered pairs from a given an
algebraic function Develop an algebraic function
given a table of ordered pairs Given a relevant
authentic problem develop and an graph an
algebraic function
Abstract
Abstract
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Sequence of Instruction
1.
2.
4.
3.
89
1.
2.
4.
3.
1. Teach prerequisite skills thoroughly.
f(x)2.54x
What are the prerequisite skills students need to
master to introduce functions and coordinate
geometry?
90
1.
2.
4.
3.
2. Teach easier skills and strategies before
more difficult ones.
Simple functions e.g. y3x
Ordered pairs that do not require complex
operations
Using numbers 0-9 then extending to gt10
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1.
2.
4.
3.
3. Introduce strategies one at a time until
mastered. Separate strategies that are
potentially confusing.
As you teach students to solve functions, when
should you introduce graphing?
92
1.
2.
4.
3.
3.
93
1.
2.
4.
3.
4. Introduce new skills, strategies, and
applications over a period of time through a
series of lessons beginning with lots of
teacher modeling, guided practice,
integration, independent practice, and review.
94
Introduction to the Concept of Functions
Input 2
Output 6
95
Functions with increasingly complex operations
y x12
96
Functions to Ordered Pairs Ordered Pairs
to Functions
97
Math Lesson Planning for Graphing Functions
Day 3 problems
2 focus
1 focus
3 focus, 3 ord. pairs
Strategy Integration
Model Strategy
Guide Strategy
Ex (3 - focus 3 discrim.) y x f(x) 2
x f(x) 3x 3 (graphing sets of ordered
pairs)
Examples (1 - focus only) y 2x
3
Ex (2 - focus only) y 7 x f(x)
4x - 1
98
Problem Solving
Problem solving is the selection and application
of known concepts or skills in a new or different
setting. For example When measuring a board of
lumber, what concepts and skills are involved?
99
Reasoning/Problem Solving
Commutative Property of Addition
Knowledge Forms
Equality
Number families
100
Complex Strategies
Divergent
Rule Relationships
Knowledge Forms
Basic Concepts
Convergent (Conventions)
Facts/Associations
101
Solving Problems or Teaching through Problem
Solving?

• Problem Solving
• Places the focus on students sense making.
• Develops the belief in students that they are
  mathematically capable.
• Provides ongoing assessment data that can be
  used for instructional decisions.
• Develops mathematical power.
• Allows an entry point for a wide range of
  students.

102
Three Part Plan for Problem Solving Instruction

• Get students mentally ready to solve
• the problem. (Preteach)
• Be sure expectations are clear.

Before
103
Generic Problem Solving Strategy
Check for accuracy
Find/calculate the solution
Plan a strategy to solve the problem
Analyze the problem
Read the problem
104
Something Else?
105
Number Family Strategy
106
(No Transcript)
107
See examples on pages in Problem Solving Chapter
(Stein et al.)
Matt had some money. Then he lost 14 dollars. Now
he has 2 dollars. How many dollars did he have
before he lost those dollars?
108
Three types of information you can provide
students during problem solving

• Conventions (facts, symbols, reminders of rules)
• Alternative methods
• Clarification of student work

109

• Get students mentally ready to solve
• the problem. (Preteach)
• Be sure expectations are clear.

Before
110
Promoting Mathematical Discourse
Why do you think your solution is correct and
makes sense?
How did you solve the problem?
Why did you solve it that way?
111
Sample Problem
Miss Spider is hosting a tea party for her 3
insect friends. If she wants each friend to have
two cookies with their tea, how many cookies will
she need to make?
112
Possible Solution Strategies
113
Moving Back to Instruction

• Children enter school with a base of math
  knowledge and the ability to interact with number
  and quantity. Very context dependent. (6)
• Instruction in math is based on the interactions
  between student, teacher, and content. Students
  must link informal knowledge with formal often
  abstract knowledge.(9)

114
What to do

• Three year grant to develop and refine
  Kindergarten math curriculum
• Y1 Intervention development and refinement
  Measurement refinement and validation
• Y2 Intervention efficacy Implementation
  analysis and hypothesis development
• Y3 Hypothesis testing re differential
  effectiveness of intervention

115
Designing Interventions

• What to do?
• Few teachers have instructional strategies in
  mathematics (Ma, 1999)
• Lack of evidence regarding effects of mathematics
  reforms (Heibert Wearne, 1993)
• Few experimental studies examining specific
  instructional practices (Gersten, Chard, Baker,
  2000)

116
Curriculum content

• Scope and sequence based on 4 integrate strands
• Numbers and operation
• Geometry
• Measurement
• Vocabulary

117
Curriculum Content (cont.)

• Key goals
• Building conceptual understanding to abstract
  reasoning via mathematical models
• Building math related vocabulary
• Procedural fluency/automaticity
• Building competence in problem solving

118
Structure

• Lessons sequenced in sets of 5
• Designed for whole class delivery in 20 minutes
• Culminates with group problem solving activity
  which integrates math discourse with strands
  taught during the previous 4 lessons